Technical Report: Full House Model in R

Introduction

We will go through examples of how our Full House Model era-adjusts batting averages (BA) using the parametric method, and bWAR (Baseball-Reference Win Above Replacement), fWAR (Fangraphs Win Above Replacement), HR (Home Run), BB (Walks) for batters, bWAR, fWAR, ERA (Earned Run Average), SO (Strikeout) for pitchers using the nonparametric method. Also we provide details about how the tables and figures in the paper and supplementary materials are produced.

We first load in relevant software packages and the data. The necessary data are collected from Baseball-Reference, Fangraphs and Github Chadwick. The details of data collection are in the Supplement Materials. The revelant datasets that involved in this technical report can be found in the Tech_report_data

For batters, each row of data consists of a player ID, a year ID, a name, a age, a league ID (lgID), a recorded number of at bats (AB), game(G), plate appearances (PA), a park-factored batting average (BA), a walk (BB), a observed hits (obs_hits), observed home run (obs_HR), a hit-by-pitch (HBP), a sacrifice bunt (SH), a sacrifice fly (SF), a park-factored hits (H), a park-factored home run (HR), a baseball-reference WAR (bWAR), a fangraph WAR (fWAR) and a size of the talent pool (pops).

For pitchers, each row of data consists of a player ID, a year ID, a name, a age, a league ID (lgID), a team ID (teamID), a recorded number of inning pitched (IP) and a game(G), a observed earned run (obs_ER), a observed home run (obs_HR), a observed hits (obs_H), a strikeout (SO), a park-factored earned run (ER), a park-factored home run (HR_PF), a park-factored hits (H_PF), a walk (BB), a hit-by-pitch (HBP), a baseball-reference WAR (bWAR), a fangraph WAR (fWAR) and a size of the talent pool (pops).

rm(list=ls())
library(tidyverse)
library(orderstats)
library(Pareto)
library(doParallel)
library(splines)
library(retrosheet)
library(kableExtra)
library(Lahman)
ncores <- detectCores() - 1
years <- 1871:2023
load("pop_data.RData")
load("bat_dat.RData")
load("pit_dat.RData")
source('source.R')

In this analysis we assume that BA \(\stackrel{i i d}{\sim} N\left(\mu_{i}, \sigma_{i}\right)\) and that talent scores \(\stackrel{i i d}{\sim} \operatorname{Pareto}(\alpha)\) where \(\alpha=1.16\). We do not have any assumptions on the distribution of the baseball statistics. This choice of alpha corresponds to the Pareto principle which is casually referred to as the \(80 / 20\) rule.

Batters

In this section, our Full House Model era-adjusts batting statistics, such as bWAR, fWAR, HR, BB and BA using the nonparametric distribution measuring the components. We also apply our Full House Model to era-adjuste BA using the parametric distribution measuring the components.

bWAR

We now try out the bWAR for batters and first we select the full-time batters. We declare the median PA after screening out individuals who batted fewer than 75 PA as our cutoff for full-time batters.

batters <- bat_dat %>% select(yearID, playerID, lgID, name, age, PA, G, bWAR, pops)
cutoff <- do.call(rbind, mclapply(years, mc.cores = ncores, FUN = function(xx){
    m <- batters %>% filter(yearID == xx) %>% filter(PA >= 75)
    data.frame(thres = median(m$PA), yearID = xx)
  }))
batters <- merge(batters, cutoff, by = 'yearID')
batters <- batters %>% mutate(comp =  bWAR / G, full_time = ifelse(PA >= thres, 'Y', 'N'))

In the 1994 and 1995, hitters and pitchers played fewer games per season than the regular season. To deal with extreme statistics from the small sample size, we motivated a shrinkage method that adjust the raw statistics toward a global average. Our shrinkage method follows a shrinkage of ballpark effect estimates motivated from Michael Schell’s book, Baseball’s All-Time Best Sluggers. This methods involves weighted average of raw components and league average, which is

\[\textrm{adjusted component} = (\textrm{raw components} \times \textrm{total ABs} + 4000 \times \textrm{league average})/(\textrm{totals AB + 4000}) \] When we use the shrinkage method to analyze our seasonal data, the shrinkage factor \(4000 / (\textrm{totals AB + 4000})\) get adjusted based on fraction of 4000 to the totals AB of the MLB teams. For example, \[\textrm{adjusted bWAR per game} = (\textrm{raw bWAR per game} \times \textrm{game} + 7 \times \textrm{league average})/(\textrm{game + 7}) \]

batters_schell <- do.call(rbind, mclapply(years, mc.cores = ncores, FUN = function(xx){
    int<- batters %>% filter(yearID == xx)
    lg_avg <- sum(int$bWAR)/sum(int$G)
    int %>% mutate(comp = (bWAR + lg_avg * 7)/(G + 7))
  }))

The following script computes the talent scores for bWAR per game from all batters and all seasons (1871 to 2022).

batters_talent_bWAR <- do.call(rbind, mclapply(years, mc.cores = ncores, function(yy){
  talent_computing_nonpara(dataset = batters_schell, component_name = "bWAR", year = yy, ystar = thresh_fun(component = batters_schell %>% filter(full_time == 'Y', yearID == yy) %>% select(comp), component_name = 'bWAR'), alpha = 1.16) })) %>% arrange(-WAR_talent)

We built a common mapping environment with regard to the provided season size based on the no-strike seasons rather than utilizing one specific season from 1871 to 2022 as the projected season. To acquire era-adjusted statistics, we project the players’ talent to this common mapping environment. Micheal Schell inspired us to use the National League seasons from 1977 through 1989, with the exception of the 1981 strike season, as the common mapping environment. The number of teams in these seasons that we choose remains constant in order to prevent the Major League’s expansion effect. We then construct an isotonic regression model of the corresponding components on the ordered talent scores. In the common mapping environment, this model provides the association between the components and the talent scores.

We set the total number of full-time players throughout all seasons to be equal to the number of components in the common mapping environment. This is due to the fact that the more components in the common mapping environment, the \(\widetilde{F}_{Y_i}(t)\) and \(F_{Y_i}(t)\) are more closely. Then we select this number of components based on the quantile mapping and the quantiles of the components are equally spaced. The talent scores of the components we choose from the common mapping environment are calculated using the isotonic regression model. Now we construct the common mapping environment completely.

no_strike <- batters_talent_bWAR %>% 
    filter(yearID < 1990, yearID >= 1977, yearID != 1981, 
           full_time == 'Y', lgID == 'NL') %>%
    arrange(WAR_talent)
  
t <- isoreg(no_strike$WAR_talent, no_strike$comp)
talent_new <- quantile(no_strike$WAR_talent, probs = (seq(0,250)/250))
comp_new <- as.stepfun(t)(talent_new)

The figure below shows the relationship between bWAR talent and bWAR per game. The black dots represent the observations from the full-time batters from the 1977 season to 1989 season with the exception of the 1981 strike season. The red dots represent the observations from the common mapping environment that we define above. Based on the figure above, the observations from the common mapping environment accurately depict the relationship between bWAR talent and bWAR per game from the 1977 season to the 1989 season with the exception of the 1981 strike season.

mapping_envir <- data.frame(talent_new = talent_new, comp_new = comp_new)
ggplot(data = no_strike, aes(x = WAR_talent, y = comp)) + geom_point() + 
  geom_point(data = mapping_envir, aes(x = talent_new, y = comp_new), color = 'red') +
  labs(x = 'bWAR talent', y = 'bWAR per game') +
  scale_x_continuous(trans='log')

ystar <- thresh_fun(comp_new, component_name = 'bWAR')
pop_new <- round(mean(no_strike$pops))
component_name = 'bWAR'
yy <- sort(comp_new)
n <- length(yy)
ytilde <- rep(0, n + 1)
if (component_name == 'bWAR' | component_name == 'fWAR') {
    ytilde[1] <- yy[1] - (yy[2] - yy[1])
}
if (component_name == 'HR' | component_name == 'BB') {
    # since the minimal HR is greater or equal to 0.
    ytilde[1] <- 0
}
ytilde[n+1] <- yy[n] + ystar
ytilde[2:n] <- unlist(lapply(2:n, function(j){
    (yy[j]+yy[j-1])/2 
}))

We now extend the method to compute hypothetical careers in the common mapping environment and obtain the era-adjusted statistics.

career_kAB_1st <- do.call(rbind, mclapply(1:30000, function(zz){
    int <- career_talent_nonpara(dataset = batters_talent_bWAR, component_name = 'bWAR', 
                         snippet = batters_talent_bWAR[zz,], alpha = 1.16)
    int
  }, mc.cores = ncores))
  
  
career_kAB_2nd <- do.call(rbind, mclapply(30001:60000, function(zz){
    int <- career_talent_nonpara(dataset = batters_talent_bWAR, component_name = 'bWAR', 
                         snippet = batters_talent_bWAR[zz,], alpha = 1.16)
    int
  }, mc.cores = ncores)) 
  
career_kAB_3rd <- do.call(rbind, mclapply(60001:nrow(batters_talent_bWAR), function(zz){
    int <- career_talent_nonpara(dataset = batters_talent_bWAR, component_name = 'bWAR', 
                         snippet = batters_talent_bWAR[zz,], alpha = 1.16)
    int
  }, mc.cores = ncores)) 
  
career_kAB <- rbind(career_kAB_1st, career_kAB_2nd, career_kAB_3rd)

Instead of using the raw G and PA in the data set, we calculate the mapped G by applying quantile mapping for the full-time hitters and non-full-time hitters separately. Quantile mapping is based on that a pth percentile player’s games in one year is equal to a pth percentile player’s games in the common mapping environment.

We also specify that worst performance of bWAR for the full time players in each season -2.

## mapping statistics
  
mapped_quan_b_raw <- do.call(rbind, mclapply(years, function(xx){
    batters_full <- batters %>% filter(yearID == xx) %>% 
      filter(full_time == 'Y') %>% arrange(-G)
    batters_less <- batters %>% filter(yearID == xx) %>% 
      filter(full_time == 'N') %>% arrange(-G)
    
    n1 <- nrow(batters_full)
    n2 <- nrow(batters_less)
    
    mapped_G_full <- c()
    mapped_G_less <- c()
    for (yy in c(1977:1980, 1982:1989)) {
      batters_ref_full <- batters %>% 
        filter(yearID == yy, full_time == 'Y') %>% arrange(-G)
      batters_ref_less <- batters %>% 
        filter(yearID == yy, full_time == 'N') %>% arrange(-G)
      n1r <- nrow(batters_ref_full)
      n2r <- nrow(batters_ref_less)
      
      mapped_G_full <- cbind(mapped_G_full, approx(x = seq((n1r-1),0)/(n1r-1), 
                                                   y = batters_ref_full$G, 
                                                   xout = seq((n1-1),0)/(n1-1))$y)
      mapped_G_less <- cbind(mapped_G_less, approx(x = seq((n2r-1),0)/(n2r-1), 
                                                   y = batters_ref_less$G, 
                                                   xout = seq((n2-1),0)/(n2-1))$y)
    }
    
    batters_full$mapped_G <- rowMeans(mapped_G_full)
    batters_less$mapped_G <- rowMeans(mapped_G_less)
    
    
    batters_full <- batters_full %>% arrange(-PA)
    batters_less <- batters_less %>% arrange(-PA)
    mapped_PA_full <- c()
    mapped_PA_less <- c()
    for (yy in c(1977:1980, 1982:1989)) {
      batters_ref_full <- batters %>% 
        filter(yearID == yy, full_time == 'Y') %>% arrange(-PA)
      batters_ref_less <- batters %>% 
        filter(yearID == yy, full_time == 'N') %>% arrange(-PA)
      n1r <- nrow(batters_ref_full)
      n2r <- nrow(batters_ref_less)
      
      mapped_PA_full <- cbind(mapped_PA_full, approx(x = seq((n1r-1),0)/(n1r-1), 
                                                     y = batters_ref_full$PA, 
                                                     xout = seq((n1-1),0)/(n1-1))$y)
      mapped_PA_less <- cbind(mapped_PA_less, approx(x = seq((n2r-1),0)/(n2r-1), 
                                                     y = batters_ref_less$PA, 
                                                     xout = seq((n2-1),0)/(n2-1))$y)
    }
    
    batters_full$mapped_PA <- rowMeans(mapped_PA_full)
    batters_less$mapped_PA <- rowMeans(mapped_PA_less)
    
    m <- rbind(batters_full, batters_less)
    data.frame(playerID = m$playerID, yearID = m$yearID,
               mapped_G_raw = round(m$mapped_G), mapped_PA_raw = round(m$mapped_PA))
    
  }, mc.cores = ncores))
   
mapped_batters_1 <- merge(career_kAB, mapped_quan_b_raw, 
                            by = c('playerID', 'yearID'))
  
min_refbWAR <- -2
  
mapped_batters_bWAR <- mapped_batters_1 %>% 
    mutate(adj_bWAR = adj_comp * mapped_G_raw) %>% 
    mutate(adj_bWAR = ifelse(adj_bWAR < min_refbWAR, min_refbWAR, adj_bWAR)) %>%
    mutate(mapped_G_bWAR = round(adj_bWAR / adj_comp))

Then we apply the same techniques for fWAR.

We also apply our full house model to BB and HR. Before we do that, we get lower bounds on walk rate and home run rate. We isolate players with batting records from the common mapping environment and restrict attention to players with 10 years of batting records. Then we obtain the 0.03rd quantile of walk rate and home run rate for the these players for seasons in which they obtained at least 400 PAs and serve the quantiles as minimum allowable rates in the common mapping environment.

IDs7789 = Batting %>% 
  filter(yearID >= 1977, yearID <= 1989, yearID != 1981) %>% 
  pull(playerID)

IDs10 = Batting %>% 
  filter(playerID %in% IDs7789) %>% 
  group_by(playerID) %>% 
  summarise(n = n()) %>% 
  filter(n >= 10) %>% 
  pull(playerID)

lowerb <- Batting %>% 
  filter(playerID %in% IDs10) %>% 
  filter(yearID >= 1977, yearID <= 1989, yearID != 1981) %>% 
  mutate(PA = AB + BB + HBP + SH + SF) %>% 
  filter(PA >= 400) %>% 
  mutate(BBrate = BB/PA, HRrate = HR/AB) %>% 
  select(playerID, yearID, BBrate, HRrate) %>% 
  summarise(Q03BB = quantile(BBrate, probs = 0.03), 
            Q03HR = quantile(HRrate, probs = 0.03))
lowerb

##        Q03BB       Q03HR
## 1 0.03512232 0.001848296

Stephen Jay Gould suggests that the BA in every season follows normal distribution and we perform Shapiro-Wilk Normality Test to verify this argument. Based on the results, 16 out of 152 seasons from 1971 to 2022 seasons fail the normality test. The 16 seasons are 1876, 1895, 1896, 1904, 1908, 1912, 1916, 1922, 1924, 1941, 1946, 1959, 1961, 1967, 1972, 1987, 1994, 1997.

batters <- bat_dat
  
  batters <- merge(batters, cutoff, by = 'yearID')
  
  batters <- batters %>% select(playerID, yearID, lgID, name, age, lgID, AB, 
                                PA, obs_hits, H, thres, bWAR, pops) %>%
    mutate(comp = ifelse(AB != 0, H / AB, 0), full_time = ifelse(PA >= thres, 'Y', 'N'))
  
  batters_schell <- do.call(rbind, mclapply(years, mc.cores = ncores, FUN = function(xx){
    int<- batters %>% filter(yearID == xx)
    lg_avg <- sum(int$H)/sum(int$AB)
    int %>% mutate(comp = (H + lg_avg * 25)/(AB + 25))
  }))
  
  normality <- do.call(rbind, mclapply(years, mc.cores = ncores, FUN = function(xx){
    m <- batters_schell %>% filter(yearID == xx, full_time == 'Y') %>% pull(comp) %>% shapiro.test()
    data.frame(yearID = xx, p_value = m$p.value)
  }))
  
  years[which(normality$p_value <= 0.05)]

##  [1] 1876 1895 1896 1904 1908 1912 1916 1922 1924 1941 1946 1959 1961 1967 1972
## [16] 1987 1994 1997 2023

Now we use both parametric and non-parametric distribution to measure the BA.

We eliminate players who had an era-adjusted bWAR or fWAR below the replacement level for more than half of their career seasons. We also set three rules to eliminate some players’ poor early and late career seasons. The three rules are

• In at least 2 consecutive seasons, the era-adjusted bWAR is below -1.5.
• In at least 2 consecutive seasons, the era-adjusted fWAR is below -1.5.
• In at least two consecutive seasons, no more than one era-adjusted bWAR or era-adjusted fWAR can be more than 0.2.

The value 0.2 is calculated from the average bWAR or fWAR of the players that disappeared from the MLB from the 1977 season to 1989 season with the exception of 1981 season.

After getting the era-adjusted statistics, we find some players’ statistics dramatically change in tails of their career, which is unrealistic in the real life. To solve it, we apply some smoothing methods to alleviate these dramatic variations, such as local polynomial regression fitting and natural cubic spline. Then natural cubic spline method has the minimal bias and is considered as the best option compared with other methods.

Also, we notice that the smoothing method could weaken player’s prime or extreme seasons. Therefore, we take the average of the smoothed era-adjusted statistics and era-adjusted statistics, which can keep player’s prime season and alleviate the dramatic changes in the tail of their career.

AVG_part <- mapped_batters_AVG_nonpara %>%
    mutate(adj_AVG = round(adj_AVG, 3)) %>% 
    select(yearID, playerID, name, adj_AVG) 
  HR_part <- mapped_batters_HR %>% 
    mutate(mapped_PA = round(mapped_PA)) %>% 
    mutate(adj_HR = round(adj_HR)) %>%
    mutate(adj_AB = round(adj_AB)) %>%
    select(yearID, playerID, adj_HR, adj_AB, HBP, SF, mapped_PA) 
  bWAR_part <- mapped_batters_bWAR %>%
    mutate(adj_bWAR = round(adj_bWAR, 2)) %>%
    select(yearID, playerID, adj_bWAR, mapped_G_bWAR) 
  fWAR_part <- mapped_batters_fWAR %>%
    mutate(adj_fWAR = round(adj_fWAR, 2)) %>%
    select(yearID, playerID, age, adj_fWAR, mapped_G_fWAR) 
  BB_part <- mapped_batters_HR %>% 
    mutate(adj_BB = round(adj_BB)) %>%
    select(yearID, playerID, BB, adj_BB)
  master_batters <- merge(BB_part, merge(AVG_part, 
                                         merge(HR_part, merge(bWAR_part, fWAR_part, 
                                                              by = c('yearID', 'playerID')), 
                                               by = c('yearID', 'playerID')), 
                                         by = c('yearID', 'playerID')), 
                          by = c('yearID', 'playerID'))
  
  master_batters <- master_batters %>% 
    mutate(adj_OBP = round((adj_AVG * adj_AB + adj_BB + HBP) / (adj_AB + adj_BB + HBP + SF), 3))
  master_batters$adj_OBP[is.na(master_batters$adj_OBP)] <- 0
  
  master_batters_nonpara <- master_batters %>% mutate(adj_BB = ifelse(adj_BB < 0, 0, adj_BB)) %>% 
    mutate(adj_AB = ifelse(mapped_PA < adj_AB, mapped_PA, adj_AB))
  master_batters_nonpara$mapped_G <- apply(master_batters_nonpara[,c(13,16)], 1, min)
  
  ## Batters
  batters <- master_batters_nonpara

  ## extract and remove bad players 
  foo <- batters %>% 
    arrange(desc(adj_AVG)) %>% 
    filter(adj_AB >= 300) %>% 
    dplyr::select(name, playerID, yearID, adj_AB, adj_AVG, adj_OBP, adj_HR, adj_fWAR, adj_bWAR) %>% 
    mutate(adj_HR_AB = round(adj_HR/adj_AB,4))
  bar <- split(foo, as.factor(foo$playerID))
  baz <- do.call(rbind, lapply(bar, function(m){
    m[which.max(m$adj_fWAR), ]
  }))
  baz <- baz %>% arrange(adj_fWAR)
  bad_players_fWAR <- baz %>% filter(adj_fWAR < 0) %>% pull(playerID)
  
  baz <- do.call(rbind, lapply(bar, function(m){
    m[which.max(m$adj_bWAR), ]
  }))
  baz <- baz %>% arrange(adj_bWAR)
  bad_players_bWAR <- baz %>% filter(adj_bWAR < 0) %>% pull(playerID)
  bad_players <- union(bad_players_bWAR, bad_players_fWAR)
  batters <- batters[!batters$playerID %in% bad_players, ]
  
  ###### investigate anomalies ######
  
  ## more on base events than PAs
  
  # check average for minimal at bats and correct issues
  batters[batters$adj_AVG > 0 & batters$adj_AB == 0, ]$adj_AVG <- 0 
  batters <- batters %>% mutate(adj_hits = round(adj_AVG * adj_AB))
  batters[batters$adj_AB > 0, ]$adj_AVG <- 
    round(batters[batters$adj_AB > 0, ]$adj_hits / batters[batters$adj_AB > 0, ]$adj_AB, 3)
  
  ## build adjusted data set
  batters_adjusted <- batters %>% 
    dplyr::select(name, playerID, age, yearID, mapped_PA, adj_AB, adj_hits, adj_HR, adj_BB, 
                  adj_AVG, adj_OBP, HBP, SF, adj_bWAR, adj_fWAR)
  colnames(batters_adjusted) <- c("name", "playerID", "age", "year", "mapped_PA", "adj_AB", "adj_H", 
                                  "adj_HR", "adj_BB", "adj_AVG", "adj_OBP", "HBP", "SF", "adj_bWAR", "adj_fWAR")
  batters_adjusted$playerID <- droplevels(as.factor(batters_adjusted$playerID))

  
  ## trim out bad players
  # first round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0)
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad == 2])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  # second round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0)
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad >= 1 & checker$len <= 2])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  # third round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    min(ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0))
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad >= 1])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  # forth round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    ifelse(m$adj_bWAR == -2, 1, 0) + ifelse(m$adj_fWAR == -2, 1, 0)
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad >= 1])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  
  ## remove tails 
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_bWAR <= 0.2, 1, 0) + ifelse(m$adj_fWAR <= 0.2, 1, 0)
    bad_tail <- sum(c(ifelse(sum(tail(bad, 2)) >= 3,1,0),
                      ifelse(sum(tail(bad, 3)) >= 5,1,0),
                      ifelse(sum(tail(bad, 4)) >= 7,1,0),
                      ifelse(sum(tail(bad, 5)) >= 9,1,0),
                      ifelse(sum(tail(bad, 6)) >= 11,1,0)))
    1:(length(bad)-bad_tail)
  })
  batters_adjusted_1 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  foo <- split(batters_adjusted_1, f = batters_adjusted_1$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_fWAR <= -1.5, 1, 0) 
    bad_tail <- sum(c(ifelse(sum(tail(bad, 2)) >= 2,1,0),
                      ifelse(sum(tail(bad, 3)) >= 3,1,0),
                      ifelse(sum(tail(bad, 4)) >= 4,1,0),
                      ifelse(sum(tail(bad, 5)) >= 5,1,0),
                      ifelse(sum(tail(bad, 6)) >= 6,1,0)))
    1:(length(bad)-bad_tail)
  })
  batters_adjusted_2 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  foo <- split(batters_adjusted_2, f = batters_adjusted_2$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_bWAR <= -1.5, 1, 0) 
    bad_tail <- sum(c(ifelse(sum(tail(bad, 2)) >= 2,1,0),
                      ifelse(sum(tail(bad, 3)) >= 3,1,0),
                      ifelse(sum(tail(bad, 4)) >= 4,1,0),
                      ifelse(sum(tail(bad, 5)) >= 5,1,0),
                      ifelse(sum(tail(bad, 6)) >= 6,1,0)))
    1:(length(bad)-bad_tail)
  })
  batters_adjusted_3 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  ## remove starts
  foo <- split(batters_adjusted_3, f = batters_adjusted_3$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0)
    bad_head <- sum(c(ifelse(sum(head(bad, 1)) == 2,1,0),
                      ifelse(sum(head(bad, 2)) >= 3,1,0),
                      ifelse(sum(head(bad, 3)) >= 5,1,0),
                      ifelse(sum(head(bad, 4)) >= 7,1,0),
                      ifelse(sum(head(bad, 5)) >= 9,1,0),
                      ifelse(sum(head(bad, 6)) >= 11,1,0)))
    if (bad_head < length(bad)) {
      (bad_head + 1):length(bad)
    }
  })
  batters_adjusted_4 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  batters_adjusted_4$playerID <- droplevels(batters_adjusted_4$playerID)
  
  # taper down average WAR for players with small PAs
  batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_fWAR <- 
    round(batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_fWAR/9,2)
  batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_bWAR <- 
    round(batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_bWAR/9,2)    
  
  batters_adjusted <- do.call(rbind, mclapply(
  split(batters_adjusted_4, f = droplevels(as.factor(batters_adjusted_4$playerID))), 
  mc.cores = ncores, FUN = function(xx){
    ## natural cubic spline
    #ns_AVG = lm(adj_AVG ~ ns(year, df=6), data=xx)
    #nn_AVG <- predict(ns_AVG, data.frame("year"= xx$year))
    #ns_HR = lm(adj_HR ~ ns(year, df=6), data=xx)
    #nn_HR <- predict(ns_HR, data.frame("year"= xx$year))
    #ns_BB = lm(adj_BB ~ ns(year, df=6), data=xx)
    #nn_BB <- predict(ns_BB, data.frame("year"= xx$year))
    #ns_bWAR = lm(adj_bWAR ~ ns(year, df=6), data=xx)
    #nn_bWAR <- predict(ns_bWAR, data.frame("year"= xx$year))
    #ns_fWAR = lm(adj_fWAR ~ ns(year, df=6), data=xx)
    #nn_fWAR <- predict(ns_fWAR, data.frame("year"= xx$year))
    
    xx %>% mutate(AVG = round(adj_AVG, 3)) %>% 
      mutate(HR = round(adj_HR)) %>%
      mutate(BB = round(adj_BB)) %>%
      mutate(ebWAR = round(adj_bWAR, 2)) %>%
      mutate(efWAR = round(adj_fWAR, 2))
  })) 
  
  bat_season <- batters_adjusted %>% 
    mutate(PA = adj_AB + BB + HBP + SF)
  bat_season[bat_season$HR >= bat_season$adj_AB, ]$HR <- 
    bat_season[bat_season$HR >= bat_season$adj_AB, ]$adj_AB

 bat_season <- bat_season %>% 
   dplyr::select(-c("adj_H", "adj_HR", "adj_BB", "adj_AVG", "adj_OBP", "adj_bWAR", "adj_fWAR", 'mapped_PA'))
 bat_season <- bat_season %>% mutate(OBP = round((AVG * adj_AB + BB + HBP)/(adj_AB + BB + HBP + SF), 3 )) %>%
   mutate(AVG = ifelse(AVG > 0, AVG, 0)) %>%
   mutate(HR = ifelse(HR > 0, HR, 0)) %>%
   mutate(BB = ifelse(BB > 0, BB, 0)) %>%
   mutate(OBP = ifelse(OBP > 0, OBP, 0))
 
 colnames(bat_season)[5] <- 'AB'
 colnames(bat_season)[8] <- 'BA'
 bat_season_nonpara <- bat_season %>% mutate(H = ceiling(AB * BA)) %>%
   mutate(BA = round(H / AB, 3)) %>%
   mutate(BA = ifelse(AB == 0, 0, BA)) %>%
   mutate(OBP = round((H+BB+HBP)/(AB+BB+HBP+SF), 3)) %>%
   mutate(OBP = ifelse(AB+BB+HBP+SF == 0, 0, OBP))
 
  bat_career_nonpara <- bat_season_nonpara %>% group_by(playerID) %>% 
    summarise(name = unique(name), 
              playerID = unique(playerID), 
              PA = sum(round(PA)), 
              AB = sum(AB), 
              H = sum(H), 
              HR = sum(round(HR)), 
              BB = sum(round(BB)), 
              BA = round(H/AB, 3), 
              HBP = sum(HBP), 
              SF = sum(SF), 
              OBP = round((H + BB + HBP)/(AB + BB + HBP + SF), 3), 
              ebWAR = sum(ebWAR), 
              efWAR = sum(efWAR)) %>% ungroup() %>% 
    arrange(desc(ebWAR))
  
  bat_career_nonpara <- bat_career_nonpara %>% mutate(BA = ifelse(AB == 0, 0, BA))
  bat_career_nonpara <- bat_career_nonpara %>% mutate(OBP = ifelse(AB + BB + HBP + SF == 0, 0, OBP))

AVG_part <- mapped_batters_AVG_para %>%
    mutate(adj_AVG = round(adj_AVG, 3)) %>% 
    select(yearID, playerID, name, adj_AVG) 
  HR_part <- mapped_batters_HR %>% 
    mutate(mapped_PA = round(mapped_PA)) %>% 
    mutate(adj_HR = round(adj_HR)) %>%
    mutate(adj_AB = round(adj_AB)) %>%
    select(yearID, playerID, adj_HR, adj_AB, HBP, SF, mapped_PA) 
  bWAR_part <- mapped_batters_bWAR %>%
    mutate(adj_bWAR = round(adj_bWAR, 2)) %>%
    select(yearID, playerID, adj_bWAR, mapped_G_bWAR) 
  fWAR_part <- mapped_batters_fWAR %>%
    mutate(adj_fWAR = round(adj_fWAR, 2)) %>%
    select(yearID, playerID, age, adj_fWAR, mapped_G_fWAR) 
  BB_part <- mapped_batters_HR %>% 
    mutate(adj_BB = round(adj_BB)) %>%
    select(yearID, playerID, BB, adj_BB)
  master_batters <- merge(BB_part, merge(AVG_part, 
                                         merge(HR_part, merge(bWAR_part, fWAR_part, 
                                                              by = c('yearID', 'playerID')), 
                                               by = c('yearID', 'playerID')), 
                                         by = c('yearID', 'playerID')), 
                          by = c('yearID', 'playerID'))
  
  master_batters <- master_batters %>% 
    mutate(adj_OBP = round((adj_AVG * adj_AB + adj_BB + HBP) / (adj_AB + adj_BB + HBP + SF), 3))
  master_batters$adj_OBP[is.na(master_batters$adj_OBP)] <- 0
  
  master_batters_para <- master_batters %>% mutate(adj_BB = ifelse(adj_BB < 0, 0, adj_BB)) %>% 
    mutate(adj_AB = ifelse(mapped_PA < adj_AB, mapped_PA, adj_AB))
  master_batters_para$mapped_G <- apply(master_batters_para[,c(13,16)], 1, min)
  
  ## Batters
  batters <- master_batters_para

  ## extract and remove bad players 
  foo <- batters %>% 
    arrange(desc(adj_AVG)) %>% 
    filter(adj_AB >= 300) %>% 
    dplyr::select(name, playerID, yearID, adj_AB, adj_AVG, adj_OBP, adj_HR, adj_fWAR, adj_bWAR) %>% 
    mutate(adj_HR_AB = round(adj_HR/adj_AB,4))
  bar <- split(foo, as.factor(foo$playerID))
  baz <- do.call(rbind, lapply(bar, function(m){
    m[which.max(m$adj_fWAR), ]
  }))
  baz <- baz %>% arrange(adj_fWAR)
  bad_players_fWAR <- baz %>% filter(adj_fWAR < 0) %>% pull(playerID)
  
  baz <- do.call(rbind, lapply(bar, function(m){
    m[which.max(m$adj_bWAR), ]
  }))
  baz <- baz %>% arrange(adj_bWAR)
  bad_players_bWAR <- baz %>% filter(adj_bWAR < 0) %>% pull(playerID)
  bad_players <- union(bad_players_bWAR, bad_players_fWAR)
  batters <- batters[!batters$playerID %in% bad_players, ]
  
  ###### investigate anomalies ######
  
  ## more on base events than PAs
  
  # check average for minimal at bats and correct issues
  batters[batters$adj_AVG > 0 & batters$adj_AB == 0, ]$adj_AVG <- 0 
  batters <- batters %>% mutate(adj_hits = round(adj_AVG * adj_AB))
  batters[batters$adj_AB > 0, ]$adj_AVG <- 
    round(batters[batters$adj_AB > 0, ]$adj_hits / batters[batters$adj_AB > 0, ]$adj_AB, 3)
  
  ## build adjusted data set
  batters_adjusted <- batters %>% 
    dplyr::select(name, playerID, age, yearID, mapped_PA, adj_AB, adj_hits, adj_HR, adj_BB, 
                  adj_AVG, adj_OBP, HBP, SF, adj_bWAR, adj_fWAR)
  colnames(batters_adjusted) <- c("name", "playerID", "age", "year", "mapped_PA", "adj_AB", "adj_H", 
                                  "adj_HR", "adj_BB", "adj_AVG", "adj_OBP", "HBP", "SF", "adj_bWAR", "adj_fWAR")
  batters_adjusted$playerID <- droplevels(as.factor(batters_adjusted$playerID))

  
  ## trim out bad players
  # first round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0)
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad == 2])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  # second round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0)
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad >= 1 & checker$len <= 2])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  # third round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    min(ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0))
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad >= 1])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  # forth round
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    ifelse(m$adj_bWAR == -2, 1, 0) + ifelse(m$adj_fWAR == -2, 1, 0)
  })
  checker <- data.frame(pid = levels(batters_adjusted$playerID), 
                        m_bad = unlist(lapply(bar, mean)), 
                        len = unlist(lapply(bar, length)))
  batters_adjusted <- batters_adjusted %>% 
    filter(!batters_adjusted$playerID %in% rownames(checker)[checker$m_bad >= 1])
  batters_adjusted$playerID <- droplevels(batters_adjusted$playerID)
  
  
  ## remove tails 
  foo <- split(batters_adjusted, f = batters_adjusted$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_bWAR <= 0.2, 1, 0) + ifelse(m$adj_fWAR <= 0.2, 1, 0)
    bad_tail <- sum(c(ifelse(sum(tail(bad, 2)) >= 3,1,0),
                      ifelse(sum(tail(bad, 3)) >= 5,1,0),
                      ifelse(sum(tail(bad, 4)) >= 7,1,0),
                      ifelse(sum(tail(bad, 5)) >= 9,1,0),
                      ifelse(sum(tail(bad, 6)) >= 11,1,0)))
    1:(length(bad)-bad_tail)
  })
  batters_adjusted_1 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  foo <- split(batters_adjusted_1, f = batters_adjusted_1$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_fWAR <= -1.5, 1, 0) 
    bad_tail <- sum(c(ifelse(sum(tail(bad, 2)) >= 2,1,0),
                      ifelse(sum(tail(bad, 3)) >= 3,1,0),
                      ifelse(sum(tail(bad, 4)) >= 4,1,0),
                      ifelse(sum(tail(bad, 5)) >= 5,1,0),
                      ifelse(sum(tail(bad, 6)) >= 6,1,0)))
    1:(length(bad)-bad_tail)
  })
  batters_adjusted_2 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  foo <- split(batters_adjusted_2, f = batters_adjusted_2$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_bWAR <= -1.5, 1, 0) 
    bad_tail <- sum(c(ifelse(sum(tail(bad, 2)) >= 2,1,0),
                      ifelse(sum(tail(bad, 3)) >= 3,1,0),
                      ifelse(sum(tail(bad, 4)) >= 4,1,0),
                      ifelse(sum(tail(bad, 5)) >= 5,1,0),
                      ifelse(sum(tail(bad, 6)) >= 6,1,0)))
    1:(length(bad)-bad_tail)
  })
  batters_adjusted_3 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  ## remove starts
  foo <- split(batters_adjusted_3, f = batters_adjusted_3$playerID)
  bar <- lapply(foo, function(m){
    bad <- ifelse(m$adj_bWAR <= 0, 1, 0) + ifelse(m$adj_fWAR <= 0, 1, 0)
    bad_head <- sum(c(ifelse(sum(head(bad, 1)) == 2,1,0),
                      ifelse(sum(head(bad, 2)) >= 3,1,0),
                      ifelse(sum(head(bad, 3)) >= 5,1,0),
                      ifelse(sum(head(bad, 4)) >= 7,1,0),
                      ifelse(sum(head(bad, 5)) >= 9,1,0),
                      ifelse(sum(head(bad, 6)) >= 11,1,0)))
    if (bad_head < length(bad)) {
      (bad_head + 1):length(bad)
    }
  })
  batters_adjusted_4 <- do.call(rbind, lapply(1:length(bar), function(j){
    foo[[j]][bar[[j]], ]
  })) %>% arrange(year)
  
  batters_adjusted_4$playerID <- droplevels(batters_adjusted_4$playerID)
  
  # taper down average WAR for players with small PAs
  batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_fWAR <- 
    round(batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_fWAR/9,2)
  batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_bWAR <- 
    round(batters_adjusted_4[batters_adjusted_4$mapped_PA <= 20, ]$adj_bWAR/9,2)    
  
  batters_adjusted <- do.call(rbind, mclapply(
  split(batters_adjusted_4, f = droplevels(as.factor(batters_adjusted_4$playerID))), 
  mc.cores = ncores, FUN = function(xx){
    ## natural cubic spline
    #ns_AVG = lm(adj_AVG ~ ns(year, df=6), data=xx)
    #nn_AVG <- predict(ns_AVG, data.frame("year"= xx$year))
    #ns_HR = lm(adj_HR ~ ns(year, df=6), data=xx)
    #nn_HR <- predict(ns_HR, data.frame("year"= xx$year))
    #ns_BB = lm(adj_BB ~ ns(year, df=6), data=xx)
    #nn_BB <- predict(ns_BB, data.frame("year"= xx$year))
    #ns_bWAR = lm(adj_bWAR ~ ns(year, df=6), data=xx)
    #nn_bWAR <- predict(ns_bWAR, data.frame("year"= xx$year))
    #ns_fWAR = lm(adj_fWAR ~ ns(year, df=6), data=xx)
    #nn_fWAR <- predict(ns_fWAR, data.frame("year"= xx$year))
    
    xx %>% mutate(AVG = round(adj_AVG, 3)) %>% 
      mutate(HR = round(adj_HR)) %>%
      mutate(BB = round(adj_BB)) %>%
      mutate(ebWAR = round(adj_bWAR, 2)) %>%
      mutate(efWAR = round(adj_fWAR, 2))
  })) 

  
  bat_season <- batters_adjusted %>% 
    mutate(PA = adj_AB + BB + HBP + SF)
  bat_season[bat_season$HR >= bat_season$adj_AB, ]$HR <- 
    bat_season[bat_season$HR >= bat_season$adj_AB, ]$adj_AB

 bat_season <- bat_season %>% 
   dplyr::select(-c("adj_H", "adj_HR", "adj_BB", "adj_AVG", "adj_OBP", "adj_bWAR", "adj_fWAR", 'mapped_PA'))
 bat_season <- bat_season %>% mutate(OBP = round((AVG * adj_AB + BB + HBP)/(adj_AB + BB + HBP + SF), 3 )) %>%
   mutate(AVG = ifelse(AVG > 0, AVG, 0)) %>%
   mutate(HR = ifelse(HR > 0, HR, 0)) %>%
   mutate(BB = ifelse(BB > 0, BB, 0)) %>%
   mutate(OBP = ifelse(OBP > 0, OBP, 0))
 
 colnames(bat_season)[5] <- 'AB'
 colnames(bat_season)[8] <- 'BA'
 bat_season_para <- bat_season %>% mutate(H = ceiling(AB * BA)) %>%
   mutate(BA = round(H / AB, 3)) %>%
   mutate(BA = ifelse(AB == 0, 0, BA)) %>%
   mutate(OBP = round((H+BB+HBP)/(AB+BB+HBP+SF), 3)) %>%
   mutate(OBP = ifelse(AB+BB+HBP+SF == 0, 0, OBP))
 
  bat_career_para <- bat_season_para %>% group_by(playerID) %>% 
    summarise(name = unique(name), 
              playerID = unique(playerID), 
              PA = sum(round(PA)), 
              AB = sum(AB), 
              H = sum(H), 
              HR = sum(round(HR)), 
              BB = sum(round(BB)), 
              BA = round(H/AB, 3), 
              HBP = sum(HBP), 
              SF = sum(SF), 
              OBP = round((H + BB + HBP)/(AB + BB + HBP + SF), 3), 
              ebWAR = sum(ebWAR), 
              efWAR = sum(efWAR)) %>% ungroup() %>% 
    arrange(desc(ebWAR))
  
  bat_career_para <- bat_career_para %>% mutate(BA = ifelse(AB == 0, 0, BA))
  bat_career_para <- bat_career_para %>% mutate(OBP = ifelse(AB + BB + HBP + SF == 0, 0, OBP))

Reulsts:

batters using non-parametric distribution

The career results for batters uses non-parametric distribution measuring the BA with top 15 bWAR batting leaders.

playerID	name	PA	AB	H	HR	BB	BA	HBP	SF	OBP	ebWAR	efWAR
bondsba01	Barry Bonds	12740	10170	3049	654	2373	0.300	106	91	0.434	153.89	145.24
mayswi01	Willie Mays	12814	11151	3475	577	1528	0.312	44	91	0.394	144.08	135.39
aaronha01	Henry Aaron	14113	12540	3887	689	1420	0.310	32	121	0.378	135.60	128.05
ruthba01	Babe Ruth	10829	8884	2672	702	1902	0.301	43	0	0.426	127.29	120.44
rodrial01	Alex Rodriguez	11917	10293	3047	547	1338	0.296	176	110	0.383	120.29	110.30
musiast01	Stan Musial	13036	11420	3579	492	1510	0.313	53	53	0.394	119.51	113.03
cobbty01	Ty Cobb	12721	11659	3726	247	971	0.320	91	0	0.376	114.48	108.77
pujolal01	Albert Pujols	13201	11432	3522	662	1523	0.308	123	123	0.391	111.86	97.34
schmimi01	Mike Schmidt	10310	8517	2331	561	1606	0.274	79	108	0.390	109.58	106.41
henderi01	Rickey Henderson	13760	11331	3195	286	2264	0.282	98	67	0.404	109.08	103.90
willite01	Ted Williams	10184	8258	2593	503	1867	0.314	39	20	0.442	107.86	107.75
speaktr01	Tris Speaker	12003	10723	3172	195	1179	0.296	101	0	0.371	102.26	95.13
morgajo02	Joe Morgan	11522	9444	2648	308	1942	0.280	40	96	0.402	100.17	96.07
robinfr02	Frank Robinson	11945	10213	3001	535	1432	0.294	198	102	0.388	99.93	95.92
ottme01	Mel Ott	11058	9370	2657	418	1624	0.284	64	0	0.393	99.74	95.72

batters using parametric distribution

The career results for batters uses parametric distribution measuring the BA with top 15 bWAR batting leaders.

playerID	name	PA	AB	H	HR	BB	BA	HBP	SF	OBP	ebWAR	efWAR
bondsba01	Barry Bonds	12740	10170	3044	654	2373	0.299	106	91	0.434	153.89	145.24
mayswi01	Willie Mays	12814	11151	3457	577	1528	0.310	44	91	0.392	144.08	135.39
aaronha01	Henry Aaron	14113	12540	3923	689	1420	0.313	32	121	0.381	135.60	128.05
ruthba01	Babe Ruth	10829	8884	2692	702	1902	0.303	43	0	0.428	127.29	120.44
rodrial01	Alex Rodriguez	11917	10293	3039	547	1338	0.295	176	110	0.382	120.29	110.30
musiast01	Stan Musial	13036	11420	3589	492	1510	0.314	53	53	0.395	119.51	113.03
cobbty01	Ty Cobb	12721	11659	3876	247	971	0.332	91	0	0.388	114.48	108.77
pujolal01	Albert Pujols	13201	11432	3510	662	1523	0.307	123	123	0.391	111.86	97.34
schmimi01	Mike Schmidt	10310	8517	2339	561	1606	0.275	79	108	0.390	109.58	106.41
henderi01	Rickey Henderson	13760	11331	3180	286	2264	0.281	98	67	0.403	109.08	103.90
willite01	Ted Williams	10184	8258	2596	503	1867	0.314	39	20	0.442	107.86	107.75
speaktr01	Tris Speaker	12003	10723	3223	195	1179	0.301	101	0	0.375	102.26	95.13
morgajo02	Joe Morgan	11522	9444	2652	308	1942	0.281	40	96	0.402	100.17	96.07
robinfr02	Frank Robinson	11945	10213	3010	535	1432	0.295	198	102	0.388	99.93	95.92
ottme01	Mel Ott	11058	9370	2662	418	1624	0.284	64	0	0.393	99.74	95.72

We apply our Full House Model to the bWAR, fWAR, SO, ERA for pitchers using the non-parametric distribution to measure the components.

pitchers

The career results for pitchers with top 15 bWAR pitching leaders.

playerID	name	IP	ER	ERA	K	ebWAR	efWAR
clemero02	Roger Clemens	5456	1701	2.81	4752	145.88	141.25
maddugr01	Greg Maddux	5646	1739	2.77	3473	113.66	120.73
johnsra05	Randy Johnson	4724	1521	2.90	5136	110.81	109.77
seaveto01	Tom Seaver	4587	1482	2.91	3656	104.31	90.78
grovele01	Lefty Grove	3518	1085	2.78	2826	102.54	98.80
verlaju01	Justin Verlander	4107	1283	2.81	3297	100.23	95.07
blylebe01	Bert Blyleven	4877	1721	3.18	3785	97.69	101.82
niekrph01	Phil Niekro	5082	1776	3.15	3364	94.37	77.47
kershcl01	Clayton Kershaw	3484	941	2.43	2980	93.78	88.83
johnswa01	Walter Johnson	4791	1766	3.32	3888	91.53	91.80
spahnwa01	Warren Spahn	5112	1832	3.23	2955	91.20	72.30
scherma01	Max Scherzer	3658	1150	2.83	3506	90.63	82.14
greinza01	Zack Greinke	4278	1433	3.01	2916	90.23	80.28
perryga01	Gaylord Perry	4977	1768	3.20	3366	89.50	94.45
carltst01	Steve Carlton	4816	1641	3.07	4221	88.70	100.34

Top 25 greatest baseball players from different scources.

rank	ebWAR	efWAR	bWAR	fWAR	ESPN	Hall of Stats
1	Barry Bonds	Barry Bonds	Babe Ruth	Babe Ruth	Babe Ruth	Babe Ruth
2	Roger Clemens	Roger Clemens	Walter Johnson	Barry Bonds	Willie Mays	Barry Bonds
3	Willie Mays	Willie Mays	Cy Young	Willie Mays	Hank Aaron	Walter Johnson
4	Babe Ruth	Henry Aaron	Barry Bonds	Ty Cobb	Ty Cobb	Willie Mays
5	Henry Aaron	Greg Maddux	Willie Mays	Honus Wagner	Ted Williams	Cy Young
6	Alex Rodriguez	Babe Ruth	Ty Cobb	Hank Aaron	Lou Gehrig	Ty Cobb
7	Stan Musial	Stan Musial	Hank Aaron	Roger Clemens	Mickey Mantle	Hank Aaron
8	Ty Cobb	Alex Rodriguez	Roger Clements	Cy Young	Barry Bonds	Roger Clemens
9	Greg Maddux	Randy Johnson	Tris Speaker	Tris Speaker	Walter Johnson	Rogers Hornsby
10	Albert Pujols	Ty Cobb	Honus Wagner	Ted Williams	Stan Musial	Houns Wagner
11	Randy Johnson	Nolan Ryan	Stan Musial	Rogers Hornsby	Pedro Martinez	Tris Speaker
12	Mike Schmidt	Ted Williams	Rogers Hornsby	Stan Musial	Honus Wagner	Ted Williams
13	Rickey Henderson	Mike Schmidt	Eddie Collins	Eddie Collins	Ken Griffey Jr.	Stan Musial
14	Ted Williams	Rickey Henderson	Ted Williams	Walter Johnson	Greg Maddux	Eddie Collins
15	Tom Seaver	Bert Blyleven	Pete Alexander	Greg Maddux	Mike Trout	Pete Alexander
16	Lefty Grove	Steve Carlton	Alex Rodrigues	Lou Gehrig	Joe DiMaggio	Alex Rodriguez
17	Tris Speaker	Lefty Grove	Kid Nichols	Alex Rodriguez	Roger Clemens	Lou Gehrig
18	Justin Verlander	Albert Pujols	Lou Gehrig	Mickey Mantle	Mike Schmidt	Mickey Mantle
19	Joe Morgan	Joe Morgan	Rickey Herderson	Mel Ott	Frank Robinson	Lefty Grove
20	Frank Robinson	Frank Robinson	Mel Ott	Randy Johnson	Rogeres Hornsby	Mel Ott
21	Mel Ott	Mel Ott	Mickey Mantle	Nolan Ryan	Cy Young	Rickey Henderson
22	Bert Blyleven	Tris Speaker	Tom Seaver	Mike Schmidt	Tom Seaver	Kid Nichols
23	Cal Ripken Jr	Justin Verlander	Frank Robinson	Rickey Henderson	Rickey Henderson	Mike Schmidt
24	Rogers Hornsby	Gaylord Perry	Nap Lajole	Frank Robinson	Randy Johnson	Nap Lajoie
25	Lou Gehrig	Rogers Hornsby	Mike Schmidt	Bert Blyleven	Christy Mathewson	Christy Mathewson

BA rankings from different sources.

rank	Peak in Full House	Career in Full House	Schell Method 1	Schell Method 2	Era-bridging Method	Raw Career
1	Rod Carew	Tony Gwynn	Tony Gwynn	Tony Gwynn	Ty Cobb	Ty Cobb
2	Ichiro Suzuki	Rod Carew	Ty Cobb	Ty Cobb	Tony Gwynn	Rogers Hornsby
3	Jose Altuve	Jose Altuve	Rod Carew	Rod Carew	Ted Williams	Shoeless Joe Jackson
4	Albert Pujols	Ichiro Suzuki	Shoeless Joe Jackson	Rogers Hornsby	Wade Boggs	Lefty O’Doul
5	Joe Mauer	Miguel Cabrera	Rogers Hornsby	Stan Musial	Rod Carew	Ed Delahanty
6	Josh Hamilton	Roberto Clemente	Ted Williams	Nap Lajoie	Shoeless Jos Jackson	Tris Speaker
7	Miguel Cabrera	Ty Cobb	Honus Wagner	Shoeless Joe Jackson	Nap Lajoie	Bill Hamilton
8	Trea Turner	Joe DiMaggio	Stan Musial	Honus Wagner	Stan Musial	Ted Williams
9	Harry Walker	Wade Boggs	Wade Boggs	Ted Williams	Frank Thomas	Dan Brouthers
10	Jeff McNeil	Buster Posey	Nap Lajoie	Wade Boggs	Ed Delahanty	Babe Ruth
11	Tony Gwynn	Mike Trout	Tris Speaker	Pete Browning	Tris Speaker	Dave Orr
12	John Olerud	Freddie Freeman	Pete Browning	Tris Speaker	Rogers Hornsby	Harry Heilmann
13	José Reyes	Joe Mauer	Willie Mays	Mike Piazza	Hank Aaron	Pete Browning
14	Alex Rodriguez	Ted Williams	Dan Brouthers	Dan Brouthers	Alex Rodriguez	Willie Keeler
15	Keith Hernandez	Stan Musial	Kirby Puckett	Tip O’Neill	Pete Rose	Bill Terry
16	Mookie Betts	Willie Mays	Babe Ruth	Kirby Puckett	Honus Wagner	Lou Gehrig
17	Pete Rose	Bill Terry	Tip O’Neill	Tony Oliva	Roberto Clements	George Sisler
18	Dee Strange Gordon	Robinson Canó	Willie Keeler	Vladimir Guerrero	George Brett	Jesse Burkett
19	Edgar Martinez	Henry Aaron	Joe DiMaggio	Mike Donlin	Don Mattingsly	Tony Gwynn
20	Stan Musial	Matty Alou	Tony Oliva	Willie Keeler	Kirby Puckett	Nap Lajoie
21	Ken Griffey	Vladimir Guerrero	Jesse Burkett	Edgar Martinez	Mike Piazza	Jake Stenzel
22	Willie McGee	Derek Jeter	Eddie Collins	Henry Aaron	Eddie Collins	Riggs Stephenson
23	Luis Arraez	Al Oliver	George Sisler	Derek Aaron	Edgar Martinez	Al Simmons
24	Robin Yount	Lou Gehrig	Lou Gehrig	Joe DiMaggio	Paul Molitor	Cap Anson
25	Derrek Lee	Edgar Martinez	Don Mattingly	Babe Ruth	Willie Mays	John McGraw

HR rankings from different sources.

rank	Peak in Full House	Career in Full House	Era-bridging Method	PPS detrending method	Peak in Schell	Career in Schell	Raw AB per HR
1	Babe Ruth	Babe Ruth	Mark McGwire	Babe Ruth	Barry Bonds	Babe Ruth	Mark McGwire
2	Willie Stargell	Mark McGwire	Juan Gonzalez	Mel Ott	Babe Ruth	Mark McGwire	Babe Ruth
3	Willie Mays	Giancarlo Stanton	Babe Ruth	Lou Gehrig	Mark McGwire	Ted Williams	Barry Bonds
4	Aaron Judge	Dave Kingman	Dave Kingman	Jimmie Foxx	Buck Frecman	Barry Bonds	Jim Thome
5	Giancarlo Stanton	Ralph Kiner	Mike Schmidt	Hank Aaron	Ed Delahanty	Mike Schmidt	Ralph Kiner
6	José Bautista	Mike Schmidt	Harmon Killebrew	Rogers Hornsby	Tim Jordan	Lou Gehrig	Harmon Killebrew
7	Mark McGwire	Willie Stargell	Frank Thomas	Cy Williams	Willie Stargell	Harmon Killebrew	Sammy Sosa
8	Chris Davis	Barry Bonds	Jose Canseco	Barry Bonds	Rogers Hornsby	Jimmie Foxx	Ted Williams
9	Luke Voit	Jimmie Foxx	Ron Kittle	Willie Mays	Jim Thome	Dave Kingman	Manny Ramirez
10	Ted Williams	Mike Trout	Willie Stargell	Ted Williams	Dave Kingman	Reggic Jackson	Adam Dunn
11	Eddie Mathews	Ted Williams	Willie McCovey	Reggie Jackson	Roy Sievers	Bill Nicholson	Ryan Howard
12	Khris Davis	David Ortiz	Darryl Strawberry	Mike Schmidt	Jeff Bagwell	Mickey Mantle	Juan Gonzalez
13	Bryce Harper	Willie McCovey	Bo Jackson	Frank Robinson	Ted Williams	Ralph Kiner	Dave Kingman
14	Mike Schmidt	Harmon Killebrew	Ted Williams	Harmon Killebrew	Kevin Mitchell	Joe DiMaggio	Russell Branyan
15	David Ortiz	Mickey Mantle	Ralph Kiner	Gavvy Cravath	Mike Schmidt	Willie Stargell	Mickey Mantle
16	Kevin Mitchell	Hank Greenberg	Pat Seerey	Honus Wagner	Lou Gehrig	Hack Wilson	Alex Rodriguez
17	Albert Pujols	Darryl Strawberry	Reggie Jackson	Willie McCovey	Fred Dunlap	Rogers Hornsby	Jimmie Foxx
18	Mickey Mantle	Jose Canseco	Ken Griffey	Harry Stovey	Harry Stovey	Darryl Strawberry	Mike Schmidt
19	Dave Kingman	Lou Gehrig	Albert Belle	Ken Griffey Jr.	Charlie Hickman	Willie McCovey	Jose Canseco
20	Gorman Thomas	Jim Thome	Dick Allen	Stan Musial	Bill Nicholson	Glenn Davis	Albert Belle
21	George Foster	Eddie Mathews	Barry Bonds	Willie Stargell	Boog Powell	Wally Berger	Khris Davis
22	Johnny Bench	Reggie Jackson	Dean Palmer	Eddie Murray	Joe DiMaggio	Eddie Mathews	Ron Kittle
23	Darrell Evans	Ryan Howard	Hank Aaron	Mark McGwire	Eddie Mathews	Harry Stovey	Carlos Delgado
24	Andruw Jones	Albert Pujols	Jimmie Foxx	Mickey Mantle	Mickey Mantle	Frank Howard	Ken Griffey Jr.
25	Reggie Jackson	Hank Sauer	Mike Piazza	Al Simmons	Tris Speaker	Mel Ott	Hank Greenberg

OBP rankings from different sources.

rank	Career Full House	Schell Method	Raw Career
1	Ted Williams	Ted Williams	Ted Williams
2	Mike Trout	Babe Ruth	Babe Ruth
3	Barry Bonds	Rogers Hornsby	John McGraw
4	Joey Votto	Barry Bonds	Billy Hamilton
5	Babe Ruth	John McGraw	Lou Gehrig
6	Mickey Mantle	Billy Hamilton	Barry Bonds
7	Bryce Harper	Topsy Hartsel	Bill Joyce
8	Lou Gehrig	Mel Ott	Jud Wilson
9	Frank Thomas	Roy Thomas	Rogers Hornsby
10	Freddie Freeman	Mickey Mantle	Ty Cobb
11	Edgar Martinez	Wade Boggs	Jimmie Foxx
12	Lance Berkman	Frank Thomas	Tris Speaker
13	Paul Goldschmidt	Lou Gehrig	Eddie Collins
14	Wade Boggs	Rickey Henderson	Ferris Fain
15	Rickey Henderson	Stan Musial	Dan Brouthers
16	Jason Giambi	Edgar Martinez	Max Bishop
17	Joe Mauer	Ty Cobb	Shoeless Joe Jackson
18	Miguel Cabrera	Dan Brouthers	Mickey Mantle
19	Joe Morgan	Tris Speaker	Mickey Cochrane
20	Prince Fielder	Joe Cunningham	Frank Thomas
21	Brian Giles	George Gore	Edgar Martinez
22	Mike Hargrove	Eddie Collins	Turkey Stearnes
23	Manny Ramirez	Ross Youngs	Stan Musial
24	Jeff Bagwell	Mike Hargrove	Cupid Childs
25	Jim Thome	Jeff Bagwell	Wade Boggs

Table 1: The percent of seasons in which BA failed the Shapiro-Wilk test based on different p-value thresholds in the Supplementary materials.

p_value	prop_season
0.05	0.15
0.10	0.24
0.20	0.34
0.30	0.39

Histogram of the p-values from Shapiro-Wilk test of normality on the BA distribution in each season in the Supplementary materials.

Table 1: Top 25 era-adjusted BA, OBP, HR, bWAR and fWAR leaders in the manuscript

rank	name	BA	name	OBP	name	HR
1	Tony Gwynn	0.342	Ted Williams	0.442	Babe Ruth	702
2	Rod Carew	0.329	Mike Trout	0.438	Henry Aaron	689
3	Jose Altuve	0.327	Barry Bonds	0.434	Albert Pujols	662
4	Ichiro Suzuki	0.327	Joey Votto	0.433	Barry Bonds	654
5	Miguel Cabrera	0.32	Babe Ruth	0.426	Reggie Jackson	578
6	Roberto Clemente	0.32	Mickey Mantle	0.42	Willie Mays	577
7	Ty Cobb	0.32	Bryce Harper	0.417	Mike Schmidt	561
8	Joe DiMaggio	0.318	Lou Gehrig	0.415	Alex Rodriguez	547
9	Wade Boggs	0.316	Frank Thomas	0.411	Frank Robinson	535
10	Buster Posey	0.316	Freddie Freeman	0.41	Ken Griffey Jr	528
11	Mike Trout	0.315	Edgar Martinez	0.41	Willie Stargell	528
12	Freddie Freeman	0.314	Lance Berkman	0.407	David Ortiz	521
13	Joe Mauer	0.314	Paul Goldschmidt	0.407	Willie McCovey	515
14	Ted Williams	0.314	Wade Boggs	0.405	Harmon Killebrew	508
15	Stan Musial	0.313	Christian Yelich	0.405	Ted Williams	503
16	Willie Mays	0.312	Rickey Henderson	0.404	Mickey Mantle	502
17	Bill Terry	0.312	Jason Giambi	0.403	Eddie Mathews	502
18	Robinson Canó	0.311	Joe Mauer	0.403	Eddie Murray	498
19	Henry Aaron	0.31	Miguel Cabrera	0.402	Jimmie Foxx	493
20	Matty Alou	0.31	Joe Morgan	0.402	Stan Musial	492
21	Vladimir Guerrero	0.31	Prince Fielder	0.4	Dave Winfield	491
22	Derek Jeter	0.31	Brian Giles	0.4	Mark McGwire	489
23	Al Oliver	0.31	Mike Hargrove	0.4	Jim Thome	484
24	Lou Gehrig	0.309	Manny Ramirez	0.4	Miguel Cabrera	480
25	Edgar Martinez	0.309	Jeff Bagwell	0.399	Lou Gehrig	479
pre-1950 in top 10	2		3		1
pre-1950 in top 25	6		3		5
proportion before 1950	0.259		0.259		0.259
talent pool	1871-2012		1871-2012		1871-2012
chance in top 10	1 in 1.29		1 in 1.99		1 in 1.05
chance in top 10	1 in 1.51		1 in 1.03		1 in 1.23

rank	name	ebWAR	name	efWAR
1	Barry Bonds	153.89	Barry Bonds	145.24
2	Willie Mays	144.08	Willie Mays	135.39
3	Henry Aaron	135.6	Henry Aaron	128.05
4	Babe Ruth	127.29	Babe Ruth	120.44
5	Alex Rodriguez	120.29	Stan Musial	113.03
6	Stan Musial	119.51	Alex Rodriguez	110.3
7	Ty Cobb	114.48	Ty Cobb	108.77
8	Albert Pujols	111.86	Ted Williams	107.75
9	Mike Schmidt	109.58	Mike Schmidt	106.41
10	Rickey Henderson	109.08	Rickey Henderson	103.9
11	Ted Williams	107.86	Albert Pujols	97.34
12	Tris Speaker	102.26	Joe Morgan	96.07
13	Joe Morgan	100.17	Frank Robinson	95.92
14	Frank Robinson	99.93	Mel Ott	95.72
15	Mel Ott	99.74	Tris Speaker	95.13
16	Cal Ripken Jr	97.39	Rogers Hornsby	94.42
17	Rogers Hornsby	97.01	Mickey Mantle	94.3
18	Lou Gehrig	95.87	Cal Ripken Jr	93.24
19	Mickey Mantle	95.37	Lou Gehrig	92.98
20	Carl Yastrzemski	95.2	Carl Yastrzemski	92.64
21	Adrián Beltré	95.01	Honus Wagner	89.81
22	Wade Boggs	92.51	Wade Boggs	87.91
23	Roberto Clemente	91.37	Mike Trout	87.87
24	Eddie Collins	90.94	Eddie Mathews	86.38
25	Mike Trout	90.43	Adrián Beltré	86.34
pre-1950 in top 10	3		4
pre-1950 in top 25	9		9
proportion before 1950	0.263		0.263
chance in top 10	1 in 1.95		1 in 3.92
chance in top 10	1 in 5.31		1 in 5.31

Table 2: Top 25 era-adjusted IP, ERA, SO. bWAR and fWAR leaders in the manuscript.

rank	name	IP	name	ERA	name	SO
1	Greg Maddux	5646	Clayton Kershaw	2.43	Nolan Ryan	6026
2	Roger Clemens	5456	Pedro Martinez	2.61	Randy Johnson	5136
3	Nolan Ryan	5319	Greg Maddux	2.77	Roger Clemens	4752
4	Warren Spahn	5112	Lefty Grove	2.78	Steve Carlton	4221
5	Phil Niekro	5082	Roger Clemens	2.81	Walter Johnson	3888
6	Don Sutton	5071	Justin Verlander	2.81	Bert Blyleven	3785
7	Gaylord Perry	4977	Max Scherzer	2.83	Tom Seaver	3656
8	Tom Glavine	4917	Roy Halladay	2.85	Don Sutton	3575
9	Bert Blyleven	4877	Randy Johnson	2.9	Max Scherzer	3506
10	Steve Carlton	4816	Tom Seaver	2.91	Greg Maddux	3473
11	Walter Johnson	4791	Cole Hamels	2.94	Gaylord Perry	3366
12	Randy Johnson	4724	Carl Hubbell	2.94	Phil Niekro	3364
13	Cy Young	4663	Whitey Ford	2.96	Justin Verlander	3297
14	Tom Seaver	4587	Curt Schilling	2.96	Pedro Martinez	3113
15	Tommy John	4585	John Smoltz	2.96	John Smoltz	3106
16	Jamie Moyer	4585	Bob Gibson	2.97	Bob Feller	3104
17	Robin Roberts	4435	Jim Palmer	2.97	Fergie Jenkins	3088
18	Pete Alexander	4356	Zack Greinke	3.01	Curt Schilling	3036
19	Zack Greinke	4278	Tim Hudson	3.03	Clayton Kershaw	2980
20	Early Wynn	4277	Juan Marichal	3.05	CC Sabathia	2960
21	Fergie Jenkins	4199	Steve Carlton	3.07	Warren Spahn	2955
22	Dennis Martinez	4182	Tom Glavine	3.1	Zack Greinke	2916
23	CC Sabathia	4169	Félix Hernández	3.1	Frank Tanana	2849
24	Justin Verlander	4107	Kevin Brown	3.12	Bob Gibson	2836
25	Jack Morris	4039	Adam Wainwright	3.12	Lefty Grove	2826
pre-1950 in top 10	1		1		1
pre-1950 in top 25	6		2		4
proportion before 1950	0.298		0.28		0.28
talent pool	1871-2006		1871-2012		1871-2006
chance in top 10	1 in 1.03		1 in 1.04		1 in 1.04
chance in top 10	1 in 1.25		1 in 1		1 in 1.05

rank	name	ebWAR	name	efWAR
1	Roger Clemens	145.88	Roger Clemens	141.25
2	Greg Maddux	113.66	Greg Maddux	120.73
3	Randy Johnson	110.81	Randy Johnson	109.77
4	Tom Seaver	104.31	Nolan Ryan	108.3
5	Lefty Grove	102.54	Bert Blyleven	101.82
6	Justin Verlander	100.23	Steve Carlton	100.34
7	Bert Blyleven	97.69	Lefty Grove	98.8
8	Phil Niekro	94.37	Justin Verlander	95.07
9	Clayton Kershaw	93.78	Gaylord Perry	94.45
10	Walter Johnson	91.53	Walter Johnson	91.8
11	Warren Spahn	91.2	Cy Young	91.28
12	Max Scherzer	90.63	Tom Seaver	90.78
13	Zack Greinke	90.23	Clayton Kershaw	88.83
14	Gaylord Perry	89.5	Don Sutton	82.98
15	Steve Carlton	88.7	Max Scherzer	82.14
16	Pedro Martinez	87.2	Pedro Martinez	82.13
17	Nolan Ryan	86.85	Zack Greinke	80.28
18	Mike Mussina	84.62	Mike Mussina	80.08
19	Curt Schilling	82.09	John Smoltz	79.36
20	Tom Glavine	81.89	Pete Alexander	77.55
21	Robin Roberts	79.01	Phil Niekro	77.47
22	Fergie Jenkins	77.83	Curt Schilling	77
23	Bob Gibson	77	Bob Gibson	76.45
24	Roy Halladay	76.07	Fergie Jenkins	75.97
25	CC Sabathia	74.5	Tommy John	75.8
pre-1950 in top 10	2		2
pre-1950 in top 25	4		4
proportion before 1950	0.28		0.28
chance in top 10	1 in 1.22		1 in 1.22
chance in top 10	1 in 1.05		1 in 1.05

Table 3: Top 25 careers according to era-adjusted bWAR (ebWAR), era-adjusted fWAR (efWAR), era-adjusted JAWS computed using bWAR (ebJAWS), and era-adjusted JAWS computed using fWAR (efJAWS) in the manuscript

name	ebWAR	name	ebJAWS	name	efWAR	name	efJAWS
Barry Bonds	153.89	Barry Bonds	109.14	Barry Bonds	145.24	Roger Clemens	103.54
Roger Clemens	145.88	Roger Clemens	107.47	Roger Clemens	141.25	Barry Bonds	103.21
Willie Mays	144.08	Willie Mays	105.16	Willie Mays	135.39	Willie Mays	98.72
Babe Ruth	137.98	Babe Ruth	100.70	Henry Aaron	128.05	Babe Ruth	90.37
Henry Aaron	135.60	Henry Aaron	95.11	Greg Maddux	120.73	Henry Aaron	89.52
Alex Rodriguez	120.29	Alex Rodriguez	91.66	Babe Ruth	120.28	Greg Maddux	88.51
Stan Musial	119.51	Stan Musial	88.38	Stan Musial	113.03	Randy Johnson	85.78
Ty Cobb	114.48	Randy Johnson	88.20	Alex Rodriguez	110.30	Alex Rodriguez	84.35
Greg Maddux	113.66	Albert Pujols	86.33	Randy Johnson	109.77	Stan Musial	83.61
Albert Pujols	111.86	Greg Maddux	85.60	Ty Cobb	108.77	Ted Williams	82.72
Randy Johnson	110.81	Mike Schmidt	84.52	Nolan Ryan	108.30	Mike Schmidt	82.20
Mike Schmidt	109.58	Lefty Grove	84.52	Ted Williams	107.75	Lefty Grove	79.31
Rickey Henderson	109.08	Ted Williams	83.54	Mike Schmidt	106.41	Ty Cobb	78.85
Ted Williams	107.86	Ty Cobb	82.62	Rickey Henderson	103.90	Rickey Henderson	78.83
Tom Seaver	104.31	Rickey Henderson	82.14	Bert Blyleven	101.82	Steve Carlton	78.32
Lefty Grove	102.54	Justin Verlander	80.34	Steve Carlton	100.34	Nolan Ryan	76.89
Tris Speaker	102.26	Joe Morgan	79.03	Lefty Grove	98.80	Albert Pujols	75.80
Justin Verlander	100.23	Tom Seaver	78.51	Albert Pujols	97.34	Justin Verlander	75.29
Joe Morgan	100.17	Cal Ripken Jr	77.54	Joe Morgan	96.07	Joe Morgan	75.14
Frank Robinson	99.93	Mike Trout	77.26	Frank Robinson	95.92	Bert Blyleven	75.12
Mel Ott	99.74	Rogers Hornsby	76.61	Mel Ott	95.72	Rogers Hornsby	74.27
Bert Blyleven	97.69	Lou Gehrig	75.70	Tris Speaker	95.13	Mike Trout	73.90
Cal Ripken Jr	97.39	Wade Boggs	75.61	Justin Verlander	95.07	Mickey Mantle	73.71
Rogers Hornsby	97.01	Clayton Kershaw	75.12	Gaylord Perry	94.45	Cal Ripken Jr	73.62
Lou Gehrig	95.87	Mickey Mantle	75.09	Rogers Hornsby	94.42	Lou Gehrig	73.41

Table 4: Top 25 four-year peaks by batting average and at bats per home run (minimum 2000 era-adjusted plate appearances) in the manuscript

name	year	BA	name	year	ABpHR
Jose Altuve	2014-2017	0.367	Barry Bonds	2001-2004	10.86
Tony Gwynn	1994-1997	0.366	Mark McGwire	1995-1998	11.15
Rod Carew	1974-1977	0.363	Babe Ruth	1918-1921	11.35
Miguel Cabrera	2010-2013	0.355	Giancarlo Stanton	2014-2017	11.85
Wade Boggs	1985-1988	0.353	Albert Pujols	2008-2011	12.20
Ichiro Suzuki	2001-2004	0.353	Eddie Mathews	1953-1956	12.34
Barry Bonds	2001-2004	0.352	Willie Stargell	1970-1973	12.45
Joe Mauer	2006-2009	0.350	Jose Canseco	1988-1991	12.46
Roberto Clemente	1964-1967	0.345	Mike Schmidt	1980-1983	12.57
Joe DiMaggio	1938-1941	0.345	José Bautista	2010-2013	12.68
Albert Pujols	2003-2006	0.343	Gorman Thomas	1978-1981	12.80
Don Mattingly	1984-1987	0.341	Ralph Kiner	1949-1952	12.87
Mike Piazza	1995-1998	0.341	Khris Davis	2015-2018	12.89
Willie Mays	1957-1960	0.340	Aaron Judge	2020-2023	13.25
Matty Alou	1966-1969	0.339	Ted Williams	1944-1947	13.26
Tim Anderson	2019-2022	0.338	Sammy Sosa	1998-2001	13.55
Stan Musial	1943-1946	0.338	Frank Howard	1967-1970	13.56
Rogers Hornsby	1922-1925	0.335	Mickey Mantle	1960-1963	13.66
Ted Williams	1943-1946	0.335	David Ortiz	2012-2015	13.71
Ty Cobb	1912-1915	0.334	Nelson Cruz	2017-2020	13.88
Trea Turner	2019-2022	0.334	Jimmie Foxx	1937-1940	13.91
Henry Aaron	1956-1959	0.333	Carlos Pena	2007-2010	13.91
Cecil Cooper	1980-1983	0.333	Dave Kingman	1976-1979	13.97
Freddie Freeman	2020-2023	0.333	Darryl Strawberry	1985-1988	13.98
Nap Lajoie	1901-1904	0.333	Jim Thome	2001-2004	14.01

Table 5: Comparison of the the top three era-adjusted seasons by Babe Ruth and Barry Bonds according to home run rate (AB per HR) in the manuscript

name	year	ABpHR	diff	ystar	balance	u	n	p_beta	pop	HR_talent
Babe Ruth	1919	10.95	0.0662271	0.0010814	0.9683747	0.9997320	118	0.9688654	2020530	5355122
Babe Ruth	1920	10.88	0.0508460	0.0003966	0.9846381	0.9998818	130	0.9847546	2517182	12061976
Babe Ruth	1926	10.83	0.0412598	0.0007058	0.9669191	0.9997455	130	0.9674563	3499956	8272038
Barry Bonds	2001	10.86	0.0661758	0.0014000	0.9594065	0.9998329	243	0.9602162	11200119	18901295
Barry Bonds	2002	10.83	0.0259097	0.0013214	0.9074378	0.9996206	244	0.9115765	11725596	9660871
Barry Bonds	2004	10.76	0.0373425	0.0016128	0.9204885	0.9996714	242	0.9235554	12829045	11901386

Histogram of the best 100 and 1000 players by their rookie year

Discussion

Year Effect on the Talent of the Replacement-level Batters

The year effect for bWAR from the talent perspective and the era-adjusted bWAR perspective will be covered in this section.

The two figures below show that the bWAR and fWAR talent of the replacement-level batters from 1871 to 2023 The line in the plot is the smoothed line after apply natural cubic spline method. The changing pattern is the similar for the talent of the replacement-level batters and talent pool. The seasons that deviate from the smoothed line are those associated with strikes and World War II, such as 1943 - 1946, 1981, 1994, 1995 and 2020 seasons.

Year Effect on the Era-adjusted WAR of the Hypothetical Batters from 2023

In this section we calculate the bWAR talent of a hypothetical 2023 hitter with 0 bWAR. Then, we compute his era-adjusted bWAR by mapping this hypothetical player to the other seasons from 1871 through 2023.

The figure above shows the era-adjusted bWAR values over time corresponding to a hypothetical batter with 0 bWAR in 2023 using the Full House Model. The fall in the mid-2000s corresponds to increase of talent pool. Despite the sharp decline in era-adjusted bWAR values in 1981 season, it is untrue that the 1981 batters with 0 bWAR would perform better than the 2023 batters. This is because the 1981 season was brief and a large number of batters underperformed replacement-level players. The problem of the sharp decline in era-adjusted bWAR values in the 1981 season is resolved when we look at the era-adjusted bWAR and era-adjusted bWAR per game of the hypothetical hitters from 2023 with 2 bWAR.

The two figures below shows the era-adjusted bWAR and era-adjusted bWAR per game of the hypothetical hitters from 2023 with 2 bWAR.

Then we also calculate the fWAR talent of a hypothetical 2023 hitter with 0 fWAR. Then, we compute his era-adjusted fWAR by mapping this hypothetical player to the other seasons from 1871 through 2023.

The figure above shows the era-adjusted fWAR values over time corresponding to a hypothetical batter with 0 fWAR in 2023 using the Full House Model. The fall in the mid-2000s corresponds to increase of talent pool. The problem of the sharp decline in era-adjusted fWAR values in the 1981 season can also be resolved when we look at the era-adjusted fWAR and era-adjusted fWAR per game of the hypothetical hitters from 2023 with 2 fWAR.

The two figures below show he era-adjusted fWAR and era-adjusted fWAR per game of the hypothetical hitters from 2023 with 2 fWAR.

Expansion Effect

It is actually supposed that the Major League’s expansion effect has a significant impact on how the talent scores vary over time and the magnitude of talent will be diluted with the expansion. However, after adding some hypothetical players who had poor performance in the early seasons, when there were fewer players, we discover that the real players did not much benefit from the adjustment of adding hypothetical players. Additionally, we make an effort to establish replacement player baselines throughout all seasons and adjust these baselines at the same level. However, the baselines we construct for every season are quite similar, and the talents of the players from earlier eras do not get much improved.

We also perform several simulations to examine the effect of season size. Using the same talent-generating process, we randomly generate different numbers of components from the same distribution and compare the top 1 talent score, top 50 talent scores, top 100 talent scores and top 300 talent scores. In each simulation, we generate two groups of components from the standard normal distribution, one consists 600 components and the other consists 300 components. Then we record the number of largest talents that group 1 is larger than the group 2. We run this simulation 1000 times and diplay the distribution of the result from each simulation. These graphs below show that top talent scores for various amounts of components from the same distribution are identical.

We also run another simulation to test the effect of season size. Instead of generating 600 components in the group 1, I generate 900 components and repeat the rest of the simulation. Then we have the similar results.

In this part, we compare the BA before and in the expansion seasons and see if there is a significant difference between them. The expansion seasons are collected from Wikipedia and they are 1879, 1892, 1900, 1901, 1961, 1962, 1969, 1977, 1993, and 1998. We compute the average and standard deviation of full-time batters’ BA before and in the expansion seasons and the results are shown below.

yearID	mean	sd	yearID	mean	sd
1878	0.281	0.043	1879	0.270	0.041
1891	0.264	0.027	1892	0.259	0.029
1899	0.293	0.037	1900	0.291	0.034
1900	0.291	0.034	1901	0.284	0.039
1960	0.268	0.025	1961	0.272	0.029
1961	0.272	0.029	1962	0.271	0.026
1968	0.250	0.027	1969	0.261	0.029
1976	0.264	0.030	1977	0.273	0.028
1992	0.265	0.026	1993	0.276	0.028
1997	0.276	0.027	1998	0.276	0.027

The average and standard error of the difference between the full-time batters’ BA before and in the expansion seasons are shown below.

avg	se
-9e-04	0.0024288

We also show Willie Mays’s seasonal bWAR per game in his 22 years MLB career from 1951 to 1973 season except the 1953 season. The red dots represent the seasons that the MLB were experiencing the expansion. The result show that the expansion did not have any significant positive or negative effects on Willie Mays’s prime season or the tail of his career.

We also show Randy Johnson’s seasonal bWAR per game in his 22 years MLB career from 1988 to 2009 season. The red dots represent the seasons that the MLB were experiencing the expansion. The result show that the expansion did not have any significant positive or negative effects on Randy Johnson’s prime season.

4 moments of BA

This figure below shows that four statistical moments of the batting average distribution from 1871 to 2021 season. Points in red correspond to seasons surrounding the peak of WWII (1941-1946).

Distribution Sensitity Analysis

Figure 2: Distribution sensitivity analysis in the Supplementary materials.

Table 3: Comparsion between deteriorated estimation regime and Z-scores in the Supplementary materials.

	correct Pareto dist	incorrect Pareto dist	normal dist	folded normal dist
beat or ties	1	1	1.000	1
strictly beat	1	1	0.995	1

Z-scores for 1997 Gwynn and 1911 Cobb from park factor adjusted BAs and regular unadjusted BAs.

name	year	zscore_BA	normality_BA	zscore_obs_BA	normality_obs_BA
Ty Cobb	1911	3.425657	0.1625329	3.460512	0.0436831
Tony Gwynn	1997	3.822984	0.0167243	3.339753	0.1744079

talent pool Calculation

Based on the talent pool computed from the supplementary materials, we are able to obtain the proportion of the talent pool before 1950 in different season spans. For example, we would like to calculate the proportion of the talent pool before 1950 from 1871 to 2006. Given the talent pool are evenly distribution from the age 20 to 29, the cumulative MLB-eligible males aged 20 to 29 from 1871 to 1879 is equal to 90% of the talent pool in 1871. Similarly, the cumulative MLB-eligible males aged 20 to 29 from 2000 to 2006 is equal to 60% of the talent pool in 2006. The code below shows how to compute the proportion of the talent pool before 1950 from 1871 to 2006.

MLBpops <- bat_dat %>% group_by(yearID) %>% summarise(pops = unique(pops)) %>% group_by(yearID) %>%
  summarise(population = round(mean(pops), 2))

# MLBpops contains the talent pool from 1871 to 2006
n <- MLBpops[MLBpops$yearID %in% c(1871, seq(1880, 2000, 10), 2006),]
p <- n
p$population[1] <- n$population[1] /10*9
p$population[15] <- n$population[15] /10*6
o <- p %>% mutate(population = round(population, 2)) %>% 
  mutate(cpp = round(cumsum(p$population)/(sum(p$population)), 3))

kable(o[o$yearID == 1950,c(1,3)]) %>% 
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"))

yearID	cpp
1950	0.292

Top 25 bWAR and fWAR leaders for MLB players with era-adjusted bWAR and fWAR

name	ebWAR	name	efWAR
Barry Bonds	153.89	Barry Bonds	145.24
Roger Clemens	145.88	Roger Clemens	141.25
Willie Mays	144.08	Willie Mays	135.39
Babe Ruth	137.98	Henry Aaron	128.05
Henry Aaron	135.60	Greg Maddux	120.73
Alex Rodriguez	120.29	Babe Ruth	120.28
Stan Musial	119.51	Stan Musial	113.03
Ty Cobb	114.48	Alex Rodriguez	110.30
Greg Maddux	113.66	Randy Johnson	109.77
Albert Pujols	111.86	Ty Cobb	108.77
Randy Johnson	110.81	Nolan Ryan	108.30
Mike Schmidt	109.58	Ted Williams	107.75
Rickey Henderson	109.08	Mike Schmidt	106.41
Ted Williams	107.86	Rickey Henderson	103.90
Tom Seaver	104.31	Bert Blyleven	101.82
Lefty Grove	102.54	Steve Carlton	100.34
Tris Speaker	102.26	Lefty Grove	98.80
Justin Verlander	100.23	Albert Pujols	97.34
Joe Morgan	100.17	Joe Morgan	96.07
Frank Robinson	99.93	Frank Robinson	95.92
Mel Ott	99.74	Mel Ott	95.72
Bert Blyleven	97.69	Tris Speaker	95.13
Cal Ripken Jr	97.39	Justin Verlander	95.07
Rogers Hornsby	97.01	Gaylord Perry	94.45
Lou Gehrig	95.87	Rogers Hornsby	94.42

Sensitivity Analysis for the condition where some talented potential baseball players fail to start their sports career in baseball.

In this section, we perform a sensitivity analysis for the condition where some talented potential baseball players fail to start their sports career in baseball. For example, people argue that Kyler Murray and Pat Mahomes are playing in the NFL, but they are also considered as the two of the most talented potential baseball players. Competition from other sports is fierce at the upper end of the talent pool where multiple sport opportunities are common.

In this sensitivity analysis, we assume the 10th, 20th, …, 100th talented potential baseball players fail to start their sports career in baseball, which indicates the player with 10th largest bWAR or fWAR is paired with 11th largest talent, the player with 20th largest bWAR or fWAR is paired with 22th largest talent, and so on. Then we mapped their talents into the common mapping environment we build before and compute the era-adjusted bWAR and fWAR. We perform this analysis for the seasons after 1950 season based on the effect of baseball integration. We also perform this analysis for the seasons after 1994 season based on the effect of MLB strike.

The tables below are the top 25 ebWAR and efWAR for batters and pitchers combined with respect to this sensitivity analysis.

without rm	ebWAR	rm after 1950	ebWAR	rm after 1994	ebWAR
Barry Bonds	153.89	Barry Bonds	153.89	Barry Bonds	153.89
Roger Clemens	145.88	Roger Clemens	145.88	Roger Clemens	145.88
Willie Mays	144.08	Willie Mays	144.08	Willie Mays	144.08
Babe Ruth	137.98	Babe Ruth	137.98	Babe Ruth	137.98
Henry Aaron	135.60	Henry Aaron	135.60	Henry Aaron	135.60
Alex Rodriguez	120.29	Alex Rodriguez	120.29	Alex Rodriguez	120.29
Stan Musial	119.51	Stan Musial	119.51	Stan Musial	119.51
Ty Cobb	114.48	Ty Cobb	114.48	Ty Cobb	114.48
Greg Maddux	113.66	Greg Maddux	113.66	Greg Maddux	113.66
Albert Pujols	111.86	Albert Pujols	111.86	Albert Pujols	111.86
Randy Johnson	110.81	Randy Johnson	110.81	Randy Johnson	110.81
Mike Schmidt	109.58	Mike Schmidt	109.58	Mike Schmidt	109.58
Rickey Henderson	109.08	Rickey Henderson	109.08	Rickey Henderson	109.08
Ted Williams	107.86	Ted Williams	107.86	Ted Williams	107.86
Tom Seaver	104.31	Tom Seaver	104.31	Tom Seaver	104.31
Lefty Grove	102.54	Lefty Grove	102.54	Lefty Grove	102.54
Tris Speaker	102.26	Tris Speaker	102.26	Tris Speaker	102.26
Justin Verlander	100.23	Justin Verlander	100.23	Justin Verlander	100.23
Joe Morgan	100.17	Joe Morgan	100.17	Joe Morgan	100.17
Frank Robinson	99.93	Frank Robinson	99.93	Frank Robinson	99.93
Mel Ott	99.74	Mel Ott	99.74	Mel Ott	99.74
Bert Blyleven	97.69	Bert Blyleven	97.69	Bert Blyleven	97.69
Cal Ripken Jr	97.39	Cal Ripken Jr	97.39	Cal Ripken Jr	97.39
Rogers Hornsby	97.01	Rogers Hornsby	97.01	Rogers Hornsby	97.01
Lou Gehrig	95.87	Lou Gehrig	95.87	Lou Gehrig	95.87

without rm	efWAR	rm after 1950	efWAR	rm after 1994	efWAR
Barry Bonds	145.24	Barry Bonds	145.24	Barry Bonds	145.24
Roger Clemens	141.25	Roger Clemens	141.25	Roger Clemens	141.25
Willie Mays	135.39	Willie Mays	135.39	Willie Mays	135.39
Henry Aaron	128.05	Henry Aaron	128.05	Henry Aaron	128.05
Greg Maddux	120.73	Greg Maddux	120.73	Greg Maddux	120.73
Babe Ruth	120.28	Babe Ruth	120.28	Babe Ruth	120.28
Stan Musial	113.03	Stan Musial	113.03	Stan Musial	113.03
Alex Rodriguez	110.30	Alex Rodriguez	110.30	Alex Rodriguez	110.30
Randy Johnson	109.77	Randy Johnson	109.77	Randy Johnson	109.77
Ty Cobb	108.77	Ty Cobb	108.77	Ty Cobb	108.77
Nolan Ryan	108.30	Nolan Ryan	108.30	Nolan Ryan	108.30
Ted Williams	107.75	Ted Williams	107.75	Ted Williams	107.75
Mike Schmidt	106.41	Mike Schmidt	106.41	Mike Schmidt	106.41
Rickey Henderson	103.90	Rickey Henderson	103.90	Rickey Henderson	103.90
Bert Blyleven	101.82	Bert Blyleven	101.82	Bert Blyleven	101.82
Steve Carlton	100.34	Steve Carlton	100.34	Steve Carlton	100.34
Lefty Grove	98.80	Lefty Grove	98.80	Lefty Grove	98.80
Albert Pujols	97.34	Albert Pujols	97.34	Albert Pujols	97.34
Joe Morgan	96.07	Joe Morgan	96.07	Joe Morgan	96.07
Frank Robinson	95.92	Frank Robinson	95.92	Frank Robinson	95.92
Mel Ott	95.72	Mel Ott	95.72	Mel Ott	95.72
Tris Speaker	95.13	Tris Speaker	95.13	Tris Speaker	95.13
Justin Verlander	95.07	Justin Verlander	95.07	Justin Verlander	95.07
Gaylord Perry	94.45	Gaylord Perry	94.45	Gaylord Perry	94.45
Rogers Hornsby	94.42	Rogers Hornsby	94.42	Rogers Hornsby	94.42

The tables below are the top 25 ebJAWS and efJAWS for batters and pitchers combined with respect to this sensitivity analysis.

without rm	ebJAWS	rm after 1950	ebJAWS	rm after 1994	ebJAWS
Barry Bonds	109.14	Barry Bonds	108.86	Barry Bonds	109.01
Roger Clemens	107.47	Roger Clemens	106.80	Roger Clemens	107.03
Willie Mays	105.16	Willie Mays	104.96	Willie Mays	105.32
Babe Ruth	100.70	Babe Ruth	100.70	Babe Ruth	100.70
Henry Aaron	95.11	Henry Aaron	94.38	Henry Aaron	95.03
Alex Rodriguez	91.66	Alex Rodriguez	91.01	Alex Rodriguez	91.01
Stan Musial	88.38	Stan Musial	87.15	Stan Musial	88.24
Randy Johnson	88.20	Randy Johnson	87.10	Randy Johnson	87.78
Albert Pujols	86.33	Lefty Grove	84.52	Greg Maddux	84.78
Greg Maddux	85.60	Mike Schmidt	84.25	Mike Schmidt	84.53
Mike Schmidt	84.52	Albert Pujols	84.14	Lefty Grove	84.52
Lefty Grove	84.52	Greg Maddux	83.77	Albert Pujols	84.14
Ted Williams	83.54	Ted Williams	83.36	Ted Williams	83.59
Ty Cobb	82.62	Ty Cobb	82.60	Ty Cobb	82.60
Rickey Henderson	82.14	Rickey Henderson	80.89	Rickey Henderson	81.05
Justin Verlander	80.19	Joe Morgan	78.01	Joe Morgan	79.04
Joe Morgan	79.03	Tom Seaver	77.36	Tom Seaver	78.49
Tom Seaver	78.50	Rogers Hornsby	76.61	Cal Ripken Jr	76.85
Cal Ripken Jr	77.54	Cal Ripken Jr	75.98	Rogers Hornsby	76.61
Mike Trout	77.26	Lou Gehrig	75.70	Lou Gehrig	75.70
Rogers Hornsby	76.61	Wade Boggs	74.72	Mickey Mantle	75.11
Lou Gehrig	75.70	Tris Speaker	74.67	Bert Blyleven	75.01
Wade Boggs	75.61	Mickey Mantle	74.50	Wade Boggs	75.00
Clayton Kershaw	75.10	Mel Ott	74.19	Tris Speaker	74.67
Mickey Mantle	75.09	Justin Verlander	73.64	Mel Ott	74.19

without rm	efJAWS	rm after 1950	efJAWS	rm after 1994	efJAWS
Roger Clemens	103.54	Roger Clemens	103.11	Roger Clemens	103.26
Barry Bonds	103.21	Barry Bonds	102.94	Barry Bonds	103.06
Willie Mays	98.72	Willie Mays	98.49	Willie Mays	98.84
Babe Ruth	90.37	Babe Ruth	90.37	Babe Ruth	90.37
Henry Aaron	89.52	Henry Aaron	88.70	Henry Aaron	89.45
Greg Maddux	88.51	Greg Maddux	87.72	Greg Maddux	88.14
Randy Johnson	85.78	Randy Johnson	85.08	Randy Johnson	85.52
Alex Rodriguez	84.35	Alex Rodriguez	83.77	Alex Rodriguez	83.77
Stan Musial	83.61	Stan Musial	82.43	Stan Musial	83.46
Ted Williams	82.72	Ted Williams	82.41	Ted Williams	82.78
Mike Schmidt	82.20	Mike Schmidt	81.85	Mike Schmidt	82.21
Lefty Grove	79.31	Lefty Grove	79.31	Lefty Grove	79.31
Ty Cobb	78.85	Ty Cobb	78.83	Ty Cobb	78.83
Rickey Henderson	78.83	Rickey Henderson	77.38	Steve Carlton	78.32
Steve Carlton	78.32	Steve Carlton	77.22	Rickey Henderson	77.72
Nolan Ryan	76.91	Nolan Ryan	75.67	Nolan Ryan	76.94
Albert Pujols	75.80	Bert Blyleven	74.31	Joe Morgan	75.14
Joe Morgan	75.14	Rogers Hornsby	74.27	Bert Blyleven	75.12
Bert Blyleven	75.12	Joe Morgan	73.99	Rogers Hornsby	74.27
Justin Verlander	74.86	Albert Pujols	73.53	Mickey Mantle	73.72
Rogers Hornsby	74.27	Lou Gehrig	73.41	Albert Pujols	73.53
Mike Trout	73.90	Mickey Mantle	73.19	Lou Gehrig	73.41
Mickey Mantle	73.71	Cal Ripken Jr	72.06	Cal Ripken Jr	72.77
Cal Ripken Jr	73.62	Walter Johnson	72.00	Walter Johnson	72.00
Lou Gehrig	73.41	Mel Ott	71.10	Mel Ott	71.10

Multiverse Analysis

Batting Average

We will test four factors in our Full House Model and use batting average to illustrate it. These four factors are park-factor effect, population change, component distribution, and season size effect. The table shows the value of the era-adjusted BA of Tony Gwynn in the 1997 season minus the era-adjusted BA of Ty Cobb in the 1911 season under different configurations. The PF column indicates whether we apply the park-factor adjustment to the BA. The YES indicates we apply park-factor adjustment, and NO indicates we did not. The pops column indicates the population changes we apply to the talent pool. The 0.5_favor shows we consider 50% favorite sport, which is the talent pool we use; The 0.75_favor shows we consider 75% favorite sport; The 1_favor shows we consider 100% favorite; The constant shows we assume the talent pool did not change over time and we set it to 1 million; The erosion shows we consider minor league erosion. The details about the how we estimate the talent pool can be found in the tech report. The para column indicates we use parametric distribution or non-parametric distribution to measure the BA in each season. The para indicates we use parametric distribution to measure BA and nonpara indicates we use non-parametric distribution to measure BA. The league column indicates we consider two different league sizes as the number of components in each season. The Historical shows we use historical league size as the number of components in each season. The Fixed shows that we compute the maximum number of components in every season and consider this value as the fixed number of components each season. The diff column shows the value of the era-adjusted BA of Tony Gwynn in the 1997 season minus the BA of Ty Cobb in the 1911 season

Table 5: The value of the era-adjuted BA of Tony Gwynn in the 1997 season minus the era-adjusted BA of Ty Cobb in the 1911 season under different configurations in the supplementary materials

PF	pops	para	league	diff
YES	0.5_favor	para	historical	0.020
YES	0.5_favor	para	fixed	0.008
YES	0.5_favor	nonpara	historical	0.040
YES	0.5_favor	nonpara	fixed	0.040
YES	0.75_favor	para	historical	0.017
YES	0.75_favor	para	fixed	0.006
YES	0.75_favor	nonpara	historical	0.035
YES	0.75_favor	nonpara	fixed	0.035
YES	1_favor	para	historical	0.014
YES	1_favor	para	fixed	0.003
YES	1_favor	nonpara	historical	0.023
YES	1_favor	nonpara	fixed	0.023
YES	constant	para	historical	0.006
YES	constant	para	fixed	-0.005
YES	constant	nonpara	historical	-0.002
YES	constant	nonpara	fixed	-0.002
YES	erosion	para	historical	0.010
YES	erosion	para	fixed	-0.001
YES	erosion	nonpara	historical	0.006
YES	erosion	nonpara	fixed	0.006
NO	0.5_favor	para	historical	0.005
NO	0.5_favor	para	fixed	0.002
NO	0.5_favor	nonpara	historical	0.029
NO	0.5_favor	nonpara	fixed	0.029
NO	0.75_favor	para	historical	0.003
NO	0.75_favor	para	fixed	-0.001
NO	0.75_favor	nonpara	historical	0.027
NO	0.75_favor	nonpara	fixed	0.027
NO	1_favor	para	historical	0.000
NO	1_favor	para	fixed	-0.003
NO	1_favor	nonpara	historical	0.015
NO	1_favor	nonpara	fixed	0.015
NO	constant	para	historical	-0.007
NO	constant	para	fixed	-0.010
NO	constant	nonpara	historical	-0.003
NO	constant	nonpara	fixed	-0.003
NO	erosion	para	historical	-0.004
NO	erosion	para	fixed	-0.008
NO	erosion	nonpara	historical	0.007
NO	erosion	nonpara	fixed	0.007

bWAR

We also use bWAR to test the effect of population change, season size and trimming method in our Full House Model. The table shows the value of the career bWAR of Willie Mays minus the career bWAR of Babe Ruth under different configurations before and after applying the trimming method. The pops column indicates the population changes we apply to the talent pool. The 0.5_favor shows we consider 50% favorite sport, which is the talent pool we use; The 0.75_favor shows we consider 75% favorite sport; The 1_favor shows we consider 100% favorite; The constant shows we assume the talent pool did not change over time and we set it to 1 million; The erosion shows we consider minor league erosion. The details about the how we estimate the talent pool can be found in the tech report. The league column indicates we consider two different league sizes as the number of components in each season. The historical shows we use historical league size as the number of components in each season. The fixed shows that we compute the maximum number of components in every season and consider this value as the fixed number of components each season. The diff_after indicates the value of the career bWAR of Willie Mays minus the career bWAR of Babe Ruth under different configurations after applying the trimming method. The diff_before indicates the value of the career bWAR of Willie Mays minus the career bWAR of Babe Ruth under different configurations before applying the trimming method.

pops	league	diff_after	diff_before
0.5_favor	historical	6.22	11.50
0.5_favor	fixed	6.21	11.52
0.75_favor	historical	1.37	6.57
0.75_favor	fixed	1.35	6.61
1_favor	historical	-2.76	-0.98
1_favor	fixed	-2.82	-0.78
constant	historical	-20.95	-21.34
constant	fixed	-20.97	-22.01
erosion	historical	-16.68	-18.93
erosion	fixed	-18.97	-19.63

Table 6: Era-adjusted JAWS (eJAWS) rankings computed with respect to talent pool estimates A-E in the manuscript

name	eJAWS	name	eJAWS	name	eJAWS	name	eJAWS	name	eJAWS
Barry Bonds	106.17	Barry Bonds	106.47	Willie Mays	106.10	Babe Ruth	112.24	Babe Ruth	117.14
Roger Clemens	105.50	Roger Clemens	104.84	Babe Ruth	105.83	Barry Bonds	106.50	Ty Cobb	108.87
Willie Mays	101.94	Willie Mays	104.02	Barry Bonds	105.22	Roger Clemens	104.36	Willie Mays	107.52
Babe Ruth	95.50	Babe Ruth	100.28	Roger Clemens	103.35	Willie Mays	104.19	Barry Bonds	106.98
Henry Aaron	92.32	Henry Aaron	94.03	Henry Aaron	97.05	Ty Cobb	100.50	Cy Young	106.41
Alex Rodriguez	88.00	Stan Musial	88.33	Ty Cobb	92.31	Walter Johnson	95.30	Roger Clemens	105.75
Greg Maddux	87.06	Alex Rodriguez	87.46	Stan Musial	91.79	Henry Aaron	93.98	Cap Anson	105.38
Randy Johnson	86.99	Greg Maddux	86.28	Lefty Grove	89.55	Stan Musial	93.94	Walter Johnson	104.81
Stan Musial	86.00	Randy Johnson	86.07	Ted Williams	87.53	Lefty Grove	92.66	Honus Wagner	101.93
Mike Schmidt	83.36	Ty Cobb	86.04	Alex Rodriguez	85.86	Honus Wagner	91.14	Henry Aaron	98.54
Ted Williams	83.13	Ted Williams	85.27	Walter Johnson	85.74	Tris Speaker	90.48	Stan Musial	98.13
Lefty Grove	81.91	Lefty Grove	85.18	Greg Maddux	84.66	Cy Young	89.53	Tris Speaker	97.73
Albert Pujols	81.06	Mike Schmidt	83.93	Mike Schmidt	84.41	Ted Williams	89.32	Lefty Grove	93.98
Ty Cobb	80.74	Rickey Henderson	79.87	Randy Johnson	84.18	Rogers Hornsby	88.20	Eddie Collins	93.82
Rickey Henderson	80.48	Albert Pujols	79.37	Rogers Hornsby	83.75	Alex Rodriguez	87.94	Ted Williams	92.09
Justin Verlander	77.82	Rogers Hornsby	79.16	Tris Speaker	82.72	Greg Maddux	86.59	Rogers Hornsby	92.03
Joe Morgan	77.09	Joe Morgan	78.39	Lou Gehrig	81.15	Randy Johnson	85.56	Nap Lajoie	88.60
Cal Ripken Jr	75.58	Walter Johnson	77.96	Joe Morgan	79.85	Eddie Collins	85.07	Greg Maddux	87.34
Mike Trout	75.58	Tris Speaker	77.37	Mel Ott	79.75	Mel Ott	84.97	Randy Johnson	86.86
Rogers Hornsby	75.44	Lou Gehrig	77.32	Honus Wagner	79.60	Lou Gehrig	84.76	Mel Ott	86.76
Bert Blyleven	75.06	Justin Verlander	76.57	Rickey Henderson	78.68	Mike Schmidt	83.19	Lou Gehrig	86.26
Lou Gehrig	74.56	Mickey Mantle	76.25	Mickey Mantle	78.68	Cap Anson	81.33	Alex Rodriguez	85.12
Mickey Mantle	74.40	Mike Trout	75.43	Tom Seaver	76.67	Albert Pujols	81.03	Pete Alexander	83.86
Steve Carlton	74.40	Bert Blyleven	75.43	Steve Carlton	76.54	Rickey Henderson	80.28	Mike Schmidt	83.62
Tom Seaver	74.15	Steve Carlton	75.33	Frank Robinson	76.46	Jimmie Foxx	78.52	Mickey Mantle	83.50

Multiverse Analysis: Home Run

We will test three factors in our Full House Model and use home run to illustrate it. These four factors are park-factor effect, population change, and season size effect. The table shows the value of the era-adjusted HR of Barry Bonds in the 2001 season minus the era-adjusted HR of Babe Ruth in the 1920 season under different configurations. The PF column indicates whether we apply the park-factor adjustment to the HR. The YES indicates we apply park-factor adjustment, and NO indicates we did not. The pops column indicates the population changes we apply to the talent pool. The 0.5_favor shows we consider 50% favorite sport, which is the talent pool we use; The 0.75_favor shows we consider 75% favorite sport; The 1_favor shows we consider 100% favorite; The constant shows we assume the talent pool did not change over time and we set it to 1 million; The erosion shows we consider minor league erosion. The details about the how we estimate the talent pool can be found in the tech report. The league column indicates we consider two different league sizes as the number of components in each season. The Historical shows we use historical league size as the number of components in each season. The Fixed shows that we compute the maximum number of components in every season and consider this value as the fixed number of components each season. The diff column shows the value of the era-adjusted HR of Barry Bonds in the 2001 season minus the era-adjusted HR of Babe Ruth in the 1920 season.

Table 6: The value of the era-adjusted AB per HR of Barry Bonds in the 2001 season minus the era-adjusted AB per HR of Babe Ruth in the 1920 season under different configurations in the supplementary materials

PF	pops	league	diff
YES	0.5_favor	historical	-0.0366554
YES	0.5_favor	fixed	-0.0366529
YES	0.75_favor	historical	-0.0115940
YES	0.75_favor	fixed	-0.0115925
YES	1_favor	historical	0.0066636
YES	1_favor	fixed	0.0066644
YES	erosion	historical	0.0196551
YES	erosion	fixed	0.0196557
YES	constant	historical	0.0284328
YES	constant	fixed	0.0284331
NO	0.5_favor	historical	-0.0099727
NO	0.5_favor	fixed	-0.0099658
NO	0.75_favor	historical	0.0378487
NO	0.75_favor	fixed	0.0378545
NO	1_favor	historical	0.0841428
NO	1_favor	fixed	0.0841461
NO	erosion	historical	0.1004012
NO	erosion	fixed	0.1004028
NO	constant	historical	0.0995427
NO	constant	fixed	0.0995433

Table 7: Top-25 eJAWS rankings after removing people from the talent pool in the manuscript

w/o rm	eJAWS	rm 1950	eJAWS	rm 1994	eJAWS
Barry Bonds	106.17	Barry Bonds	105.90	Barry Bonds	106.03
Roger Clemens	105.50	Roger Clemens	104.96	Roger Clemens	105.15
Willie Mays	101.94	Willie Mays	101.72	Willie Mays	102.08
Babe Ruth	95.50	Babe Ruth	95.50	Babe Ruth	95.50
Henry Aaron	92.32	Henry Aaron	91.54	Henry Aaron	92.24
Alex Rodriguez	88.00	Alex Rodriguez	87.39	Alex Rodriguez	87.39
Greg Maddux	87.06	Randy Johnson	86.09	Randy Johnson	86.65
Randy Johnson	86.99	Greg Maddux	85.75	Greg Maddux	86.46
Stan Musial	86.00	Stan Musial	84.79	Stan Musial	85.85
Mike Schmidt	83.36	Mike Schmidt	83.05	Mike Schmidt	83.37
Ted Williams	83.13	Ted Williams	82.88	Ted Williams	83.18
Lefty Grove	81.91	Lefty Grove	81.91	Lefty Grove	81.91
Albert Pujols	81.06	Ty Cobb	80.72	Ty Cobb	80.72
Ty Cobb	80.74	Rickey Henderson	79.13	Rickey Henderson	79.38
Rickey Henderson	80.48	Albert Pujols	78.84	Albert Pujols	78.84
Justin Verlander	77.82	Joe Morgan	76.00	Joe Morgan	77.09
Joe Morgan	77.09	Rogers Hornsby	75.44	Rogers Hornsby	75.44
Cal Ripken Jr	75.58	Lou Gehrig	74.56	Bert Blyleven	75.06
Mike Trout	75.58	Cal Ripken Jr	74.02	Cal Ripken Jr	74.81
Rogers Hornsby	75.44	Mickey Mantle	73.84	Lou Gehrig	74.56
Bert Blyleven	75.06	Bert Blyleven	73.78	Mickey Mantle	74.41
Lou Gehrig	74.56	Tom Seaver	73.05	Steve Carlton	74.40
Mickey Mantle	74.40	Steve Carlton	72.97	Tom Seaver	74.13
Steve Carlton	74.40	Wade Boggs	72.65	Wade Boggs	73.05
Tom Seaver	74.15	Mel Ott	72.64	Mel Ott	72.64

Figure 5: Estimated WAR of a 2-WAR player in 2023 in the manuscript

Figure 6: Tail probability for outlying Ruth and Bonds seasons in the manuscript

Table 4: The bWAR rankings from Full House Model using different talent generating function in the Supplementary materials.

In this section, we use different talent generating process to verify the robustness of the model.

The table below shows the top 25 bWAR players using four different talent generating process. The talent follows folded normal distribution, normal distribution, Pareto distribution with \(\alpha = 3\) and Pareto distribution with \(\alpha = 1.16\). Given that the ranking lists produced by the four separate generating processes are identical, we can say that our Full House model’s talent generating process is fairly robust.

rank	standard normal	era-adjusted bWAR	Folded normal (mu = 0, sigma = 1)	era-adjusted bWAR	Pareto with alpha = 3	era-adjusted bWAR	Pareto with alpha = 1.16	era-adjusted bWAR
1	Barry Bonds	153.93482003704	Barry Bonds	153.934820057079	Barry Bonds	153.93482003704	Barry Bonds	153.900596672323
2	Roger Clemens	145.907695934742	Roger Clemens	145.907695925218	Roger Clemens	145.907695934742	Roger Clemens	145.907695934742
3	Willie Mays	144.20526296462	Willie Mays	144.205262974907	Willie Mays	144.20526296462	Willie Mays	144.095468112426
4	Henry Aaron	135.560701275838	Henry Aaron	135.560701289904	Henry Aaron	135.560701275838	Henry Aaron	135.601861141104
5	Babe Ruth	132.702324888145	Babe Ruth	132.702324893792	Babe Ruth	132.702324888145	Babe Ruth	132.702324888145
6	Stan Musial	119.25776183457	Stan Musial	119.257761829684	Stan Musial	119.25776183457	Stan Musial	119.509994291424
7	Alex Rodriguez	119.05928887207	Alex Rodriguez	119.059288892948	Alex Rodriguez	119.05928887207	Alex Rodriguez	119.066070765468
8	Greg Maddux	113.670801140907	Greg Maddux	113.670801146666	Greg Maddux	113.670801140907	Greg Maddux	113.670801140907
9	Ty Cobb	112.000334920716	Ty Cobb	112.000334926544	Ty Cobb	112.000334920716	Ty Cobb	112.003670261108
10	Randy Johnson	110.812019546253	Randy Johnson	110.812019560192	Randy Johnson	110.812019546253	Albert Pujols	111.852368250334
11	Mike Schmidt	109.61435217199	Mike Schmidt	109.614352185456	Mike Schmidt	109.61435217199	Randy Johnson	110.812019546253
12	Albert Pujols	109.147332313036	Albert Pujols	109.147332303033	Albert Pujols	109.147332313036	Mike Schmidt	109.591837967447
13	Rickey Henderson	109.137893304437	Rickey Henderson	109.137893313238	Rickey Henderson	109.137893304437	Rickey Henderson	109.063644359715
14	Ted Williams	108.155500712145	Ted Williams	108.155500712963	Ted Williams	108.155500712145	Ted Williams	108.037777234477
15	Tom Seaver	104.28306193596	Tom Seaver	104.283061916776	Tom Seaver	104.28306193596	Tom Seaver	104.305946556062
16	Lefty Grove	101.582214503872	Lefty Grove	101.582214523742	Lefty Grove	101.582214503872	Lefty Grove	101.582214503872
17	Tris Speaker	100.72466821009	Tris Speaker	100.72466820624	Tris Speaker	100.72466821009	Tris Speaker	100.699554124141
18	Joe Morgan	100.182322428416	Joe Morgan	100.182322426815	Joe Morgan	100.182322428416	Justin Verlander	100.240996144687
19	Frank Robinson	100.036273382032	Frank Robinson	100.036273385659	Frank Robinson	100.036273382032	Joe Morgan	100.168379881699
20	Bert Blyleven	97.6905454153586	Bert Blyleven	97.6905454263732	Bert Blyleven	97.6905454153586	Frank Robinson	99.9106587464703
21	Cal Ripken Jr	97.4212949564655	Cal Ripken Jr	97.4212949308992	Cal Ripken Jr	97.4212949564655	Bert Blyleven	97.6905454153586
22	Mel Ott	96.690996783323	Mel Ott	96.6909967816042	Mel Ott	96.690996783323	Cal Ripken Jr	97.4130854322836
23	Rogers Hornsby	95.7583931883965	Rogers Hornsby	95.7583931873413	Rogers Hornsby	95.7583931883965	Mel Ott	96.716110869205
24	Lou Gehrig	95.6758386877758	Lou Gehrig	95.675838706285	Lou Gehrig	95.6758386877758	Rogers Hornsby	95.7583931883965
25	Mickey Mantle	95.4051563355476	Mickey Mantle	95.4051563322911	Mickey Mantle	95.4051563355476	Lou Gehrig	95.6758386877758

Figure 3: The relationship between bWAR talent and bWAR per game in the supplementary materials

Figure 4: The pairings of the maximum talent scores with their corresponding era-adjusted bWAR for players in the common mapping environment in the supplementary materials

Table 7: Whether or not the distribution of the batting statistics is parametric or nonparametric affects the top 25 career batting averages and top 25 four-year peaks by batting average.

name	BA	name	BA	name	year	BA	name	year	BA
Tony Gwynn	0.342	Tony Gwynn	0.338	Jose Altuve	2014-2017	0.367	Tony Gwynn	1994-1997	0.360
Rod Carew	0.329	Ty Cobb	0.332	Tony Gwynn	1994-1997	0.366	Ty Cobb	1916-1919	0.353
Jose Altuve	0.327	Rod Carew	0.324	Rod Carew	1974-1977	0.363	Wade Boggs	1985-1988	0.351
Ichiro Suzuki	0.327	Ichiro Suzuki	0.322	Miguel Cabrera	2010-2013	0.355	Rod Carew	1974-1977	0.350
Miguel Cabrera	0.320	Jose Altuve	0.320	Wade Boggs	1985-1988	0.353	Ichiro Suzuki	2001-2004	0.348
Roberto Clemente	0.320	Roberto Clemente	0.318	Ichiro Suzuki	2001-2004	0.353	Rogers Hornsby	1921-1924	0.346
Ty Cobb	0.320	Joe DiMaggio	0.318	Barry Bonds	2001-2004	0.352	Jose Altuve	2014-2017	0.345
Joe DiMaggio	0.318	Shoeless Joe Jackson	0.316	Joe Mauer	2006-2009	0.350	Mike Piazza	1995-1998	0.345
Wade Boggs	0.316	Wade Boggs	0.314	Roberto Clemente	1964-1967	0.345	Barry Bonds	2001-2004	0.344
Buster Posey	0.316	Freddie Freeman	0.314	Joe DiMaggio	1938-1941	0.345	Joe DiMaggio	1939-1942	0.343
Mike Trout	0.315	Stan Musial	0.314	Albert Pujols	2003-2006	0.343	Don Mattingly	1984-1987	0.340
Freddie Freeman	0.314	Ted Williams	0.314	Don Mattingly	1984-1987	0.341	Henry Aaron	1956-1959	0.339
Joe Mauer	0.314	Henry Aaron	0.313	Mike Piazza	1995-1998	0.341	Roberto Clemente	1969-1972	0.338
Ted Williams	0.314	Buster Posey	0.313	Willie Mays	1957-1960	0.340	Stan Musial	1943-1946	0.338
Stan Musial	0.313	Mike Trout	0.312	Matty Alou	1966-1969	0.339	Joe Mauer	2006-2009	0.337
Willie Mays	0.312	Matty Alou	0.311	Tim Anderson	2019-2022	0.338	Miguel Cabrera	2010-2013	0.336
Bill Terry	0.312	Miguel Cabrera	0.311	Stan Musial	1943-1946	0.338	Nap Lajoie	1901-1904	0.336
Robinson Canó	0.311	Robinson Canó	0.311	Rogers Hornsby	1922-1925	0.335	Albert Pujols	2003-2006	0.336
Henry Aaron	0.310	Vladimir Guerrero	0.311	Ted Williams	1943-1946	0.335	Matty Alou	1966-1969	0.335
Matty Alou	0.310	Joe Mauer	0.311	Ty Cobb	1912-1915	0.334	Honus Wagner	1905-1908	0.335
Vladimir Guerrero	0.310	Rogers Hornsby	0.310	Trea Turner	2019-2022	0.334	Tris Speaker	1913-1916	0.334
Derek Jeter	0.310	Willie Mays	0.310	Henry Aaron	1956-1959	0.333	Ted Williams	1943-1946	0.334
Al Oliver	0.310	Kirby Puckett	0.310	Cecil Cooper	1980-1983	0.333	Freddie Freeman	2020-2023	0.332
Lou Gehrig	0.309	Bill Terry	0.310	Freddie Freeman	2020-2023	0.333	Lou Gehrig	1932-1935	0.332
Edgar Martinez	0.309	Lou Gehrig	0.309	Nap Lajoie	1901-1904	0.333	Willie Mays	1957-1960	0.332

Figure 1: Histogram of the p-values from Shapiro-Wilk test of normality on the BA distribution in each season in the Supplementary materials.